11. Vector & 3D_WB-2.pdf - Best Approach

Vector & 3D

Best Approach

Manoj Chauhan Sir (IIT Delhi)

Exp. More than 13 Yearsin Top Coaching of Kota

No. 1 Faculty of Unacademy,

By Mathematics Wizard

WorkbookPattern-2

Maths IIT-JEE ‘Best Approach’ MCSIR Vector & 3D

2Get 10% Instant Discount On Unacademy Plus [Use Referral Code : MCSIRLIVE]

VECTORS

1. Co-ordinate axis and co-ordinate

planes in 3D space

xy plane, yz plane, zx plane divide rectangular

co-ordinate system into 8 parts called octants,

named as XOYZ, X'OYZ, X'OY'Z, XOY'Z,

XOYZ', X'OYZ', X'OY'Z' and XOY'Z' are

denoted by I, II, III, IV, V, VI, VII, VIII.

Octant /I II III IV V VI VII VIII

co ordinates

x

y

z

Note :

(i) If a point P has the cordinate (x, y, z) then x, y,

z are perpendicular distances from yz, xz, xy

planes with direction

(ii) Equation of plane yz, zx, xy are writen by

x = 0, y = 0, z = 0 respectively.

(iii) Any point on xy plane will be taken as (x, y, 0)

(iv) Any point on x-axis will be taken as (x, 0, 0)

Distance between 2 points :

If co-ordinate of the points A, B are

(x1, y

1, z

1), (x

2, y

2, z

2) respectively then

AB 2 2 2(x x ) (y y ) (z z )2 1 2 1 2 1

2 2 21 1 1OA (x y z ) where O is origin

Section formula :

Let point P divides segment AB in the ratio

m : n then

P = 2 1 2 1 2 1mx nx my ny mz nz

, ,m n m n m n

Note :

(i) Co-ordinate of the centroid of triangle ABC

are

1 2 3 1 2 3 1 2 3x x x y y y z z z

, ,3 3 3

(ii) Co-ordinate of the incentre of triangle ABCare

1 2 3 1 2 3 1 2 3ax bx cx ay by cy az bz cz, ,

a b c a b c a b c

where BC = a, CA = b and AB = c

Definition of vector :A quantity with magnitude and direction iscalled as vector or a directed line segment iscalled a vector

Directed line segment AB a .

Magnitude of AB

is the length of line segment

AB. Its denoted by AB | a | a

Note :

(i) | a | 0

is meaningless.

(ii) vector can move along parallel direction meanstwo parallel vector of same size in samedirection will be same.

Position vector :

Let point P(x, y, z) then OP

is called position

vector of point P. Generally its written by

r(OP)

.

note that ˆ ˆ ˆOP xi yj zk

; where ˆ ˆ ˆi, j, k are

the unit vector along x, y, z axis (unit vectoris defined ahead) where

OP

is a directed line segment whose initial

point is origin and terminal point is P.

Note that AB p.v. of B p.v. of A

2. Section Formula in vector form :

If a

& b

are the position vectors of two pointsA & B then the p.v. of a point R which dividesAB internally in the ratio m : n is given by:

mb nar

m n

If R divides AB externally in m : n then :

mb nar

m n

P.v. of mid point of AB = a b

2

.



Using section formula we can prove that

(i) p.v. of the centroid of a triangle

ABC = 3

(ii) Incentre of the a b c

a b c

and

Excentres of the are

a b c a b c a b c

; anda b c a b c a b c

(iii) Circumcentre of the

sin 2A sin 2B sin 2C

sin 2A

(iv) orthocenter of the

tan A tan B tan C

tan A

(Use the fact that distances of orthocentrefrom the vertices are 2R cosA , 2R cosB , 2RcosC andfrom the sides are2 R cosB cosC , 2R cosCcosA , 2R cosA cosB)

Q. A (1, 1, 3), B (2, 1, 2) & C (5, 2, 6)are the position vectors of the vertices of atriangle ABC. The length of the bisector of itsinternal angle at A is :

(A) 410 (B) 4103

(C) 10 (D) none

Q. P, Q have position vectors a & b

relative to

the origin 'O' & X, Y divide PQ

internally and

externally respectively in the ratio 2 : 1. Vector

XY

=

(A) 3b a

2

(B) 4a b

3

(C) 5b a

6

(D) 4b a

3

Q. Four points A(1, –1, 1) ; B(1, 3, 1) ;

C(4, 3, 1) and D(4, – 1, 1) taken in order are

the vertices of

(A) a parallelogram which is neither a rectangle

nor a rhombus

(B) rhombus

(C) an isosceles trapezium

(D) a cyclic quadrilateral.

Q. Let , & be distinct real numbers. The

points whose position vectors are

ˆ ˆ î j k ; ˆ ˆ î j k and

ˆ ˆ î j k

(A) are collinear

(B) form an equilateral triangle

(C) form a scalene triangle

(D) form a right angled triangle

Q. If the vectors

ˆ ˆ â 3i j 2 k

, ˆ ˆ ˆb i 3 j 4 k

& ˆ ˆ ˆc 4 i 2 j 6 k

constitute the sides of

a ABC, then the length of the median

bisecting the vector c is

(A) 2 (B) 14

(C) 74 (D) 6



Q. Consider the points A, B and C with position

vectors ˆ ˆ ˆ2i 3j 5k , ˆ ˆ ˆi 2 j 3k and

ˆ ˆ7i k respectively..

Statement-1 : The vector sum

A B BC C A

= 0

Statement-2 : A, B and C form the verticesof a triangle.(A) Statement-1 is true, statement-2 is true

and statement-2 is correct explanation forstatement-1.

(B) Statement-1 is true, statement-2 is trueand statement-2 is NOT the correctexplanation for statement-1.

(C) Statement-1 is true, statement-2 is false.(D) Statement-1 is false, statement-2 is true.

Direction cosine :

If a vector OP

makes angles , , withpositive direction x, y, z axes respectively, then, , are called direction angles and cos, cos , cos are called direction cosines.

z

P

O x

y

xcos

| OP | (say),

ycos m

| OP | (say),

zcos n

| OP | (say)

Note : (i) 2 + m2 + n2 = 1(ii) cos2 + cos2 + cos2 = 1(iii) sin2 + sin2 + sin2 = 2

Q. D.C. of the vector

2 2 1ˆ ˆ ˆ2i 2 j k are , ,3 3 3

Q. Find the locus of all points P for which OP

represents a vector with direction cosine cos

= 1

2(O is origin).

Q. Find the number of unit vectors with

1 1cos and cos

2 2 .

Q. Find the number of vector with direction angles = 30o and = 30o.

3. Types of vector :

Zero vector :A vector of zero magnitude i.e.which has thesame initial & terminal point, is called a ZeroVector (also called null vector).

lt is denoted by 0

.

Note : Zero vector has many properties similar to thenumber zero.A boy throwing a ball up and catching it backin his hand. the displacement of the ball is anull Vector.

Unit vector :A vector of unit magnitude in the direction of

vector a

is called unit vector along a

, and is

denoted by a . Symbolically a

aa

Q. A rigid body rotates about an axis through the

origin with an angular velocity 10 3 m/s. If

points in the direction with position vector

ˆ ˆ ˆ(i j k) , then find

.

Q. Find the number of distinct unit vectors in spaceperpendicular to a given plane.

Q. How many unit vectors to a given line in spaceare possible ?

Q. If the sum of two unit vectors is a unit vectorthen find the magnitude of their difference and

the angle between a and b .



Equal Vectors :Two vectors are said to be equal if they havethe same magnitude, same direction &represent the same physical quantity.

Free Vectors :Free vectors are those which whentransformed into space from one point toanother point without affecting their magnitudeand direction, can be considered as freevectors i.e. the physical effects produced bythem remains unaltered. e.g. displacement,velocity

Localised Vector :A vector drawn parallel to a given vector,but through a specified point as the initial point,is called a localised vector.e.g. force, different physical effect if line ofapplication is changed.

Collinear Vectors :Two vectors are said to be collinear if theirdirected line segments are parallel disregardsto their direction. Collinear vectors are alsocalled Parallel Vectors. If they have the samedirection they are named as like vectorsotherwise unlike vectors.(a, b, c are collinear)

a

cb

Symbolically, two non zero vectors a

and b

are collinear if and only if a bk

.If K > 0, like parallel vectors.

K < 0, unlike parallel vectors

Coplanar Vectors :A given number of vectors are called coplanarif their line segments lies in the same plane.Note that “Two parallel or intersectingvectors always lies in the same plane”.

Triangle law and vector addition :

AC AB BC

or AB BC CA 0

This is known as triangle law of vectoraddition.

Note that if a & b

are the 2 co-initial vectoralong 2 adjacent sides of a parallelogram, then

a b

is denoted by co-initial diagonal of theparallelogram. Then, the triangle law can betermed as Parallelogram law of vectoraddition.

If a, b

are 2 vectors in a plane then by the

triangle law, we can find a b

, a b

.

Q. ABCD is parallelogram whose diagonals meetat P. If O is a fixed point and

OA OB OC OD OP

find

Note :(i) By using triangle law, we can define

2 1 2 1 2 1ˆ ˆ ÂB (x x )i (y y ) j (z z )k

,

where A (x1, y

1,z

1) and B(x

2, y

2,z

2)

(ii) 1 1 2 2 3 3ˆ ˆ â b (a b )i (a b ) j (a b )k

& 1 1 2 2 3 3ˆ ˆ â b (a b )i (a b ) j (a b )k

where 1 2 3ˆ ˆ â a i a j a k

& 1 2 3ˆ ˆ ˆb b i b j b k

(iii) 1 2 3ˆ ˆ â a i a j a k

(this is known as

SCALAR MULTIPLICATION), where is scalar.

a

is always collinear to a

where can be

positive or negative

Note that 31 2

1 2 3

bb bb a

a a a

(iv) a b b a

, (a b) c a (b c)

Q. Findif ˆ ˆ â 2i 3j k

and

ˆ ˆ ˆb 8i j 4k

are parallel.

Q. If ˆ Â (2i 3j), ˆ ˆB (pi qj) and

ˆ ˆC (i j) are collinear, find p and q.

Q. If mid points of BC, CA, AB are D, E, F thenfind the position vector of centroid of triangleDEF. Given that when position vector of A,

B, C are ˆ î j , ˆ ˆj k and ˆ ˆk i respectively..



Q. Three co-initial vectors of magnitade a, 2a,3a meet at a point and their directions are alongthe diagonals of 3 adjacent faces of a cube.Determine their resultant.

Q. Centre of a Room(cube) is joined to 8 cornersof cube Room(cube). 3 vectors out of thusformed 8 vectors are selected at random. Findprobability that selected vectors are coplanar.

Q. Vectors ˆ ˆ â 3i j – 2k, ˆ ˆ ˆb i 3j 4k,

ˆ ˆ ˆc 4i 2 j – 6k

constitute sides of a ABC.

Then find the length of median bisecting

vector c

.

Q. ABCDE is regular pentagon.

AB AE BC CD DE AE

Then find the value of ?

Q. In triangle ABC, ˆ ˆ ÂB i 2 j 3k

and

ˆ ˆ ˆBC 4i 5 j 6k

then find the length of 3rd

side and median?

Q. In quadrilateral ABCD

prove that figureformed by joining the mid-points ofconsecutive sides is a parallelogram.

Q. In triangle ABC, AB x

and BC y then

find AM

in terms of x

and y

, where M

divides BC in 1 : 3.

A

CBM

x

y1 : 3

4. Linear combination :

A vector r

is said to be a linear combination

of vectors 1 2 na ,a ........., a

if there exist scalars

m1 , m

2 , ..... , m

n such that

1 1 2 2 n nr m a m a ....... m a

.

Linearly Independent :

A system of vectors 1 2 na ,a ,............, a

is said

to be linearly independent if

1 1 2 2 n nm a m a ...... m a 0

m1 = m

2 = ....... = m

n = 0

Note :(i) A pair of non-collinear vectors is linearly

independent.

If a, b

non zero non-collinear vectors such

that xa yb 0 x y 0

Vector a & b

are called as base vector.

(ii) A triad (set of three) of non-coplanar vectoris linearly independent.

Q. Can ˆ î j and ˆ ˆ2i 2 j be treated as the base

vectors in xy-plane ?

Q. Can ˆ î j and ˆ î j be treated as the base

vectors in xy-plane ?

Linearly Dependent :

A set of vectors 1 2 na , a , , a

is said to be

linearly dependent if there exist scalars

1 2 nm , m , , m , not all zero, such that

1 1 2 2 n nm a m a m a 0

.

Note:(i) A pair of collinear vectors is linearly

dependent.(ii) A triad of coplanar vectors is linearly

dependent.

Result -1:

If a

and b

are two non-collinear vectors, then

every vector r

coplanar with a

and b

canbe expressed in one and only one way as a

linear combination a b; and

x y x y being

scalars. Here a

and b

are termed as basevectors.

Note : Any vector r

in xy plane can be taken as

ˆ ˆr xi yj



5. Geometrical results with vectors:(i) Straight line joining the mid points of two non

parallel sides of a trapezium is parallel to theparallel sides and half of their sum.

(ii) Tetrahedron (a pyramid on a triangularbase)

(a) Lines joining the vertices of a tetrahedron tothe centroids of the opposite faces areconcurrent and this point (P) of concurrency

with position vector a b c d

g4

is

called the centre of the tetrahedron (say G).

(b) In a tetrahedron straight lines joining the midpoints of each pair of opposite edges are alsoconcurrent at the centre of the tetrahedron.

(iii) Four diagonals of any parallelopiped (A prismwhose base is a ||gm) and the join of the midpoint of each pair of opposite edges areconcurrent and are bisected at the pointconcurrency. This point is called the centre ofthe parallopiped with position vector

a b c OA OB OCi.e.

2 2

Q. Centre of the parallelopiped formed by

ˆ ˆ ˆPA i 2 j 2k

, PB 4, –3,1

,

PC 3,5, –1

is (7, 6, 2). Find position

vector of 'P'.

Q. The diagonals of the three faces of theparallopiped drawn from the same vertex areprolonged half their lengths. Show that thethree points thus obtained are coplanar withthe opposite vertex.

6. Vector Equation of a line :It is possible to express the position vectorsof points on given lines and planes in terms ofsome fixed vectors and variable scalars calledparameters, such that :

(i) For arbitrary value of the parameter, theresulting position vector represent point on thelocus.

(ii) Conversely, the position vector of each pointon the locus correspond to a definite value(s)of the parameter.Vector equation of a line passing through two

point A ( a

) & B ( b

) : r

= a

+ t( b

– a

)where t is a parameter.

If the line passes through the point A(a

) & is

parallel to the vector b

: r a tb

These two equations proves to be very usefulin vector algebra.

Note:(i) Two lines in a plane are either intersecting or

parallel conversely two intersecting or parallellines must be in the same plane.

(ii) However in space we can have two neitherparallel nor intersecting lines.Such non coplanar lines are known as skewlines. if two lines are parallel and have acommon point then they are coincident.

Q. Find the p.v. of the point of intersection of thelines

(i) ˆ ˆ ˆ ˆ ˆ ˆr i j 10k (2i 3j 8k)

ˆ ˆ ˆ ˆ ˆ ˆand r 4i 3j k (i 4 j 7k)

(ii) ˆ ˆ ˆ ˆ ˆr 3i 6 j ( 4i 3j 2k)

ˆ ˆ ˆ ˆ ˆand r 2i 7k ( 4i j k)



(iii) ˆ ˆ ˆr t(3i j k)

ˆ ˆ ˆ ˆand r 2i s( 6i 2 j 2k)

(iv) ˆ ˆ ˆ ˆr 2k (3i 2 j k)

ˆ ˆ ˆ ˆ ˆ ˆand r 3i 2j 3k (6i 4 j 2k)

Q. Through the middle point M of the sideAD of ||gm

D(d) C(b + d)

M

R

A(Origin)

B(b)

Q

ABCD, a straight line BM is drawn

intersecting AC at R and CD produced at Q.

use vectors to prove that QR = 2RB.

Q. In AOB, E is the mid point of OB and F

divides BA in the ratio 1 : 2 use vectors to

prove that OP 3

PF 2 .

P 2

FE

B

O A

b2

(2b + a)3

1

(a)

(b)

Q. In a triangle ABC, D divides BC in the ratio

3 : 2 and E divides CA in the ratio 1 : 3. The

lines AD and BE meet at H and CH meets

AB in F.

H 3

DE

C

A(origin)B

(3c + 2b)5

2

(b)

(c)

1

3

F

Find the ratio in which F divides AB.

Q. Let OACB be parallelogram with O at theorigin & OC a diagonal. Let D be the midpoint of OA. Using vector method provethat BD & CO intersect in the same ratio.Determine this ratio.

Q. In ABC, EF is drawn || to BC with E on ABand F on AC. If BF and CE meet at L.Prove that AL bisects BC.

Vector equation of the bisectors of the anglesbetween the lines :

r a b

and r a c

are

ˆ ˆr a t(b c)

and ˆ ˆr a s(b c)

Q. Use vectors to prove that the internal (external)

bisectors of a triangle divide the opposite base

internally (externally )in the ratio of the side

containing the angle.

Note : We can make use of the vector equation to

prove the concurrency of angle bisectors by

finding the point of intersection of two angle

bisectors and then satisfy the point of

intersection in the third equation.

7. Test of collinearity :Three points A, B, C with position vectors

a, b, c

respectively are collinear, if and only if

there exist scalars x, y, z not all zero

simultaneously such that :

xa yb zc 0

, where x + y + z = 0.



Q. Find whether the following points are collinearor not

(i) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2i 5j 4k; i 4 j 3k; 4i 7 j 6k

(ii) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ3i 4 j 3k; 4i 5j 6k; 4i 7 j 6k

Note : Collinearly can also be checked by firstfinding the equation of line through two pointsand satisfying the third point.

Q. Given 2sin a 2sin b c 0

if A(a), B(b), c(c)

are collinear. Find .

Q. Vectors P,Q,

act at 'O' (origin) have a

resultant R

. If any transversal cuts their lineof action at A, B, C respectively, then show

that P Q R

OA OB OC .

8. Dot (Scalar) Product

a.b | a || b | cos

for two non zero vectors

a & b

, where is angle between a

and b

and 0

(i) Useful Results :

(a) If a.b 0

is acute

a.b 0

= /2

a.b 0

is obtuse

(b) Maximum value of a.b | a || b |

(c) Minimum value of a.b | a || b |

(ii) General Expression for dot product :

If 1 2 3ˆ ˆ â a i a j a k

&

1 2 3ˆ ˆ ˆb b i b j b k

then

1 1 2 2 3 3a.b a b a b a b

&

1 1 2 2 3 3

2 2 2 2 2 21 2 3 1 2 3

a b a b a ba.bcos

| a || b | a a a b b b

(iii) Any vector a

can be expressed as

ˆ ˆ ˆ ˆ ˆ â i i a j j a k k

.

(iv) Properties of Scalar Product :

(a) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ î . i j . j k . k 1; i . j j . k k . i 0

(b) Dot product is commutative i.e. a.b b.a

(c) Conventionally 2 2a.a | a | a

(d) Dot product is distributive

a.(b c) a.b a.c

(v) Projection of a.b

a on b| b |

.

Note that vector component of a

along b

is

2

a.bb

b

and vector component of a

perpendicular to 2

a.bb a b

| b |

.

Q. If ˆ ˆ â 2i 3j 6k

and ˆ ˆ ˆb 2i 5 j 14k

If = Projection of a on b

Projection of b on a

. Find .

Q. Express the vector ˆ ˆ â 5i 2 j 5k

as the

sum of two vectors such that one is parallel to

ˆ ˆb 3i k

and the other perpendicular to b

.

(vi) Simple Identities :

(a)2 2(a b).(a b) a b

(b) 2 2 2(a b) a 2a.b b (a b).(a b)

(c)2 2 2(a b) a 2a.b b

(d) 2 2(a b) (a b) 4a.b

(e)22 2 2 2(a b c ) a b c 2 (a.b)

(f) 2 21a.b (a b) (a b)

4

Q. If ˆ ˆ â 5i j 3k

, ˆ ˆ ˆb i 3j 5k

find the

angle between a

and a 2b

.



Q. Find the angle between a b and a b

if

a 2, b 1 & a b3

.

Q. Find p q

if ˆ ˆ2p q i j

, ˆ ˆp 2q i j

.

Q. | a | 11, | b | 23

and | a b | 30

find

| a b |

.

Q. If a b c 0

, | a | 3;| b | 1

and

| c | 4

. Find (a.b)

.

Q. A vector of magnitude 25 is collinear with

ˆ ˆ â 2i 3j 4k

and makes obtuse angle with

negative z-axis find the vector.

Q. Prove that â b 2 2cos 2cos2

and â b 2 2cos 2sin2

,

where is the angle between the vectors

a & b

.

Q. Let u

be a vector on rectangular coordinate

system with sloping angle 60°. Suppose that

iu

is geometric mean of u

and i2u

where i is the unit vector along x-axis then

u

has the value equal to ba where

a, b N, find the value (a + b)3 + (a – b)3.

Q. Prove that 22

22 |b||a|

ba

b

b

a

a

Q. A point P moves in space such that

P(A) P(B) 0

then the locus of the point P

is the interior of the sphere with AB asdiameter.

Some More Illustrations on dot product :

(1) Cosine formula for triangles.

(2) Projection formula for triangles.

(3) cos ( ) = cos cos sin sin

(4) A quadrilateral whose diagonals bisect each

other at right angle is rhombous.

(5) Angle in a semicircle is a right angle.

(6) Acute angle between the diagonals of a cube.

(7) Medians to the equal sides of an isosceles

triangle are equal and converse.

(8) Concurrency of altitudes/ right bisectors/

medians/ angle bisectors of a triangle.

(9) If two circles intersect then the line joining their

centres is perpendicular to the common chord.

(10) O/G/C of a scalene triangle are collinear G

divides the line joining O and C in the ratio of

2 : 1. (O orthocentre, G centroid,

C circumcentre).

Q. Find vector equation of an ellipse/ a Hyperbola

whose foci are the position vectors c

and –c

and length of major / Transversal Axis is 2a.

Q. Arc AC of the quadrant of a circle with centre

as origin and radius unity subtends a right angle

at the origin. Point B divides the arc AC in the

ratio 1 : 2. Express the vector c

in terms of a

and b

.

Q. Two adjacent sides of a parallelogram ABCD

are given by AB

= ˆ ˆ ˆ2i 10j 11k and

ˆ ˆ ÂD i 2 j 2k

. The side AD is rotated

by an actue angle in the plane of the

parallelogram so that AD becomes AD', If AD'

makes a right angle with the side AB, then the

cosine of the angle is given by [JEE 2010]

(A) 8

9(B)

17

9

(C) 1

9(D)

4 5

9

Q. Given that ˆ â i j

and ˆ ˆ ˆb i 2 j are two

vectors. Find a unit vector coplanar with a

and b

and perpendicular to a

.



Q. If a, b, c

are coplanar vectors, prove that

a b c

a.a a.b a.c 0

b.a b.b b.c

.

Q. Find perpendicular distance of a point A (a)

from a line passing through the point b

and

parallel to c

and reflection of a point in a line.

Q. Find the foot of the perpendicular from the

origin on the line ˆ ˆ ˆ ˆ ˆr i 4k (2i j k)

.

Q. Show that the sides of a trapezium havingequal non parallel sides are equally inclined tothe parallel sides.

Q. A triangle OAB is right angled at O, squaresOALM and OBPQ are constructed on thesides OA and OB externally. Show that thelines AP and BL intersect on the altitudethrough ‘O’.

Q. If two pairs of opposite edges of a tetrahedronare at right angles, then show that the thirdpair is also at right angle. Further show thatfor such a tetrahedron, the sum of the squaresof each pair of opposite edges is the same.

Q. ABCD is a tetrahedron such that theperpendiculars AK, BL, CM and DN to theopposite faces are concurrent. Prove that

(i) Any two opposite edges of the tetrahedronare orthoganal.

(ii) K is the orthocentre of the triangle BCD.

Q. Obtain the equation to the locus of a point R

(r)

in space which is equidistant from (i) two

given points with p.v.’s a

and b

(ii) three

given points with p.v.’s a

, b

and c

AndAndinterpret it geometrically.

Q. If the perpendiculars from two vertices B andC to the opposite faces of a tetrahedron ABCDintersect, then BC is perpendicular to AD andperpendicular from A and D to the oppositefaces also intersect.

Q. Find radius of the sphere circumscribing andinradius of a regular tetrahedron whose edgeis k.

Q. If a b and c, are unit vectors, then

a b b c c a

2 2 2does NOT exceed

(A) 4 (B) 9(C) 8 (D) 6

Q. Use vectors to prove that in a ABCcos 2A + cos 2B + cos 2C > – 3/2. Also provethat the distance between the circumcentre and

the centroid is 2 2 2 21

R (a b c )9

.

9. Cross (Vector) Product :

ˆa b | a || b | sin n

where n is the unit vector perpendicular to

the plane containing the vectors a

and b

such

that a

and b

and n forms a right handedscrew system.

Q. Find equation of a line which passes through

the point with p.v. a

and perpendicular to the

lines r b p

and r c q

.

Lagranges Identity :

Since | a b |

is very frequently needed for

which Lagranges identity is very useful.

2 2 2| a b | a b (a.b)

Q. Prove that for any vector

2 2 2ˆ ˆ ˆ| a i | | a j | | a k | 2a

Properties of Cross Product :

(i) a b 0

a b(a 0; b 0)

i.e. a

and

b

are collinear / Linearly dependent.

(ii) a b b a

(not commutative).

(iii) a (b c) a b a c

(distributive to be proved later using tripleproduct).



(iv) (a b) c) a (b c)

vector product is not associative.

(v) ˆ ˆ ˆ ˆ ˆ î i j j k k 0 and

ˆ ˆ î j k; ˆ ˆ ˆj k i; ˆ ˆ ˆk i j .

Expression for :

a b

where 1 2 3ˆ ˆ â a i a j a k

and

1 2 3ˆ ˆ ˆb b i b j b k

1 2 3

1 2 3

ˆ ˆ î j k

a b a a a

b b b

Q. Find the equation of the line through the point

with p.v. ˆ ˆ2i 3j and perpendicular to the

vectors ˆ ˆ Â i 2 j 3k

and ˆ ˆ ˆB 3i 4 j 53k

.

Q. If ˆ ˆ ˆ ˆ ˆ ˆ(2i 6 j 27k) (i j k) 0 , find

and .

Geometrical Interpretation :

| a b | | a | | b | sin

denotes the area of

parallelogram whose two adjacent sides are

the vectors a

and b

.Area of a parallelogram/quadrilateral, if

diagonal vectors 1d

and 2d

are known

1 2

1A | d d |

2

Note : Area of the triangle = | a b |

2

.

Vector of magnitude r perpendicular to the

plane of a

and b

is r(a b)

| a b |

.

If is the angle between a

and b

then

| a b |sin

| a | | b |

.

Q. For a non zero vector a

, if a.b a.c

and

a b a c

. Prove that b c

.

Q. Find :

(i) A vector of magnitude 6 perpendicular to

the plane ABC(ii) Area of triangle ABC(iii) Length of altitude from A(iv) Equation of the plane ABC

A(1,–1,2)

C(0,2,1)B(2,0,–1)

Perpendicular distance of a point P from a line

using cross product.

(a b) cd

| c |

.

P(a

b

cr = b + tc

d

)

Q. Let ˆ ˆ â i 4 j 2k;

ˆ ˆ ˆb 3i 2 j 7k

and

ˆ ˆ ˆc 2i j 4k

. Find the vector d

which is

perpendicular to both a

and b

and satisfy

c.d 15 .

Q. Four vertices O, A, B, C of a tetrahedron

satisfy ˆ ˆ ÔA OB i j k;

ÔB OC i;

ˆ ÔC OA i j,

find CA CB

.

Q. If d,c,b,a

are position vectors of the vertices

of a cyclic quadrilateral ABCD prove that :

a x b bxd dxa

b a d a

bx c cxd dx b

b c d c

( ) . ( ) ( ) . ( )0



Q. Find the unknown vector R

satisfying

R B C B

and R.A 0

where

ˆ ˆA 2i k,

ˆ ˆ ˆB i j k

and

ˆ ˆ ˆC 4i 3j 7k

.

Q. If a, b, c

are the position vectors then the vector

area of ABC is 1

(c b) (a b)2

1(a b) (b c) (c a)

2

A(a )

B(b) C( )c

Note : If 3 points position vectors a, b and c

are the collinear then a b b c c a 0

.Q. Prove that 3 points with position vectors

a b, a – b

and a b

are collinear for

R.

Q. Prove the identity (a – b) (a b) 2(a b)

and give its geometrical interpretation.

Q. If c,b,a

and d

are unit vectors such that

1dc·ba

and 2

1c·a

, then

[JEE-2009]

(A) c,b,a

are non-coplanar

(B) d,c,b

are non-coplanar

(C) d,b

are non-parallel

(D) d,a

are parallel and c,b

are parallel

Q. Let OA a, OB 10a 2b and OC b

where O, A & C are non-collinear points. Let'p' denote the area of the quadrilateral OABC,and let 'q' denote the area of the parallelogramwith OA and OC as adjacent sides. If p = kq.Find k.

10. Skew Lines :In 3-D geometry the lines which do notintersect and are not parallel are called skewlines.

Shortest Distance Between 2 Skew Lines :

Method - I, Method - II

Q. Find shortest distance between

ˆ ˆ ˆ ˆ ˆr i j 2i – j k ,

ˆ ˆ ˆ ˆ ˆ ˆr 2i j – k 3i – 5j 2k

.

Q.(i) Vector equation of two lines are

ˆ ˆ ˆr 1 – t i t – 2 j 3 – 2t k,

ˆ ˆ ˆr s 1 i 2s– 1 j – 2s 1 k

Find shortest distance.

(ii) Vector equation of two lines are

ˆ ˆ ˆ ˆ ˆ ˆr i 2 j k i – j k ,

ˆ ˆ ˆ ˆ ˆ ˆr 2i – j – k 2i j 2k

Find shortest distance.

Shortest Distance Between 2 Parallel Lines :

Q. Find the distance between the lines L1 and L

2

given by

ˆ ˆ ˆ ˆ ˆ ˆr i 2 j – 4k 2i 3j 6k

ˆ ˆ ˆ ˆ ˆ ˆr 3i 3j – 5k 2i 3j 6k

Tetrahedron :

It's a pyramid with triangular base. 4 vertices,

4 surfaces, 6 edges, 1 centre, 3 pair of

opposite edges.

Centre of tetrahedron is the point of

intersection of line joining the vertices of

tetrahderon to the centroid of opposite faces.

Centre is also the point of intersection of line

joining the mid points of opposite edges.

Parallelopiped :

It is actually a prism with parallelogram base.

8 vertices, 6 faces, 12 sides, 4 body diagonals,

12 face diagonals.



Centre of Parallelopiped :

Point of intersection of body diagonals.

Q. For the tetrahedron ABCD, A(0, 1, 2),B (3, 0, 1), C (4, 3, 6), D (2, 3, 2)Find.

(i) P.v. corresponding to the point of concurrencyof the join of the mid points of each pair ofopposite edges.

(ii) P.v. of the foot N from the vertex A and theperpendicular distance of A from the faceBCD.

(iii) Image of A in plane face BCD.(iv) Altitude of tetrahedron from the vertex A.(v) Volume of tetrahedron(vi) Unit vectors normal to the plane face ABC

and ADC.(vii) Acute angle between the planes ABC and

ADC.(viii) Shortest distance between the skew lines AD

and BC and the angle between them.(ix) Equation of the plane through ABC.

11. Product of 3 or more vectors

Triple Product (a b).c

(a b) c

When 3 vectors are involved with a dot or across between them, then 6 different symbols

are : (1) (a.b)c

(2) (a.b).c

(3) (a.b) c

(4) (a b)c

(5) (a b).c

(6) (a b) c

First is scalar multiple of a vector 2, 3,4 aremeaningless 5 and 6 are scalar and vector tripleproduct respectively.

Scalar Triple Product :

ˆ(a b).c | a || b |sin n.c | a || b || c | sin cos

where = a ^ b ; n ^ c

but gm| a || b | sin area of || OACB

base area and | c | cos h

Hence (a b).c

geometrically interpret

the volume of the parallelopiped whose3 cotorminous edges are the vectors

a, b and c

.

Note :

(i) (a b).c

is also known as box product and

is written a b c

.

(ii) If the vector c

also lies in the plane of a, b

then a b c

= 0.

Hence for three vectors a, b, c

if

a b c

= 0 then a, b, c

are in the same plane

& convers is also true.(iii) Points A, B, C & D will be coplanar if

AB AC AD 0

(iv) If a, b, c

are unit vectors such that their box

procuct is unity i.e. [a b c]

= 1 then a, b, c

are mutually perpendicular to each other.

Q. If ˆ ˆ û 2i j k

, ˆ ˆ ˆv i j k

and w

is unit

vector then find the maximum value of [u v w]

.

(v) General expression for [a b c]

:

when a, b, c

are expressed in terms of ˆ ˆ î, j, k

[a b c] (a b).c

1 2 3

1 2 3 1 2 3 1 2 3

1 2 3 1 2 3

ˆ ˆ î j k a a aˆ ˆ â a a c i c j c k b b b

b b b c c c

Note : If 3 directed line segments are in the same plane

then the value of this determinant vanishes. This

can be used to determine the equation of a

plane through 3 non collinear points.

Properties of STP :

(i) Scalar triple product of three vectors when

two of them are collinear/linearly dependent

is equal to zero. (two rows identical

determinant is zero)

(ii) If the cyclic order of vector retains then the

value of the STP does not change i.e.

[a b c]

= [b c a]

= [c a b]

= [a c b]



(iii) The position of dot and cross can be

interchanged provided the cyclic order of the

vectors a, b, c

remains undisturbed.

we have (a b).c (b c).a (c a).b

Also

(a b).c c.(a b)

(b c).a a.(b c)

(c a).b b.(c a)

(iv) [a b c d ] [a c d ] [ b c d ]

(v)

For right handed system

[a b c] 0 wherea, b, c are

and for left handed system non coplanar

[a b c] 0

Q. Find the value of for which three points with

p.v.'s A (1, 0, 3), B (–1, 3, 4), C (1, 2, 1) and

D (d, 2, 5) are in the same plane. Also find

the equations of the plane through ABC.

Q. Find the value of p for which the

vectors ˆ ˆ ˆ ˆ ˆ ˆ(p 1)i 3j pk , pi (p 1) j 3k

and ˆ ˆ ˆ3i pj (p 1)k are coplanar..

Q. Find the altitude of the parallelopiped whose

coterminous edges are vectors

ˆ ˆ â i j k

, ˆ ˆ ˆb 2i 4 j k

, ˆ ˆ ˆc i j 3k

where a & b

are base vectors.

Q. The vector OP

= k2j2i turns through

a right angle, passing through the positive

x-axis on the way. Find the vector in its new

position.

Q. ˆ ˆ ˆ ˆ2v v (i 2 j) 2 j k

then find 3 v

.

Q. Show that :

ˆ ˆ ˆ ˆ ˆ â × b = [i a b]i + [j a b] j + [k a b] k

.

Q. If w,v,u

are three non-coplanar unit

vectors and , , are the angles between

vandu

, wandv

, uandw

respectively and

z,y,x

are unit vectors along the bisectors

of the angles , , respectively. Prove that

xzzyyx

.

2

sec2

sec2

secwvu16

1 2222

[JEE 2003, 4 out of 60]

Q. [a b b c c a] 2[a b c]

. This identity

can be geometrically interpreted as :

(Volume of a cuboid whose three coterminous

edges are the face diagonals of the cuboid is

twice the volume of the cuboid, whose three

coterminous edges are the vectors a, b, c

).

This is also conclusive that if a, b, c

are

coplanar then a b,b c

, c a

are also

coplanar.

Q. Prove that : [a b b c c a] 0

Q. Let ˆ ˆ ˆ ˆ ˆ â i 2 j 4k, b i j 4k

and

2ˆ ˆ ˆc 2i 4j ( 1)k

be coplanar vectors.

Then the non - zero vector a c

is :

[JEE (Main) 2019]

(A) ˆ ˆ10i 5j (B) ˆ ˆ14i 5j

(C) ˆ ˆ14i 5 j (D) ˆ ˆ10i 5 j

Q. Prove that :

.a .b .c

[ m n][a b c] m.a m.b m.c

n.a n.b n.c

l l l

l,

where , m, n l & a, b, c

are non–coplanar

vectors



Q. Prove that 2

a.a a.b a.c

[a b c] b.a b.b b.c

c.a c.b c.c

,

where a, b, c

are non–coplanar vectors.

Q. Prove that

p.a q.a a

[a b c](p q) p.b q.b b

p.c q.c c

.

Q. Prove that volume of a tetrahedron

OABC = 1 1

V [a b c] OA OB OC6 6

where O is the origin.

Q. Let the volume of a parallelopipedwhose coterminous edges are given by

ˆ ˆ ˆ ˆ ˆ û i j k, v i j 3k

and

ˆ ˆ ˆw 2i j k

be 1 cu. unit. If be the angle

between the edges u and w

, then cos can

be : [JEE Main 2020]

(A) 7

6 6(B)

5

7

(C) 7

6 3(D)

5

3 3

To express scalar triple product of three vectors

in terms of any three non coplanar vectors m l,

and n

:Let

1 2 3 1 2 3a a a m a n ;b b b m b n, l l

1 2 3c c c m c n l

then1 2 3

1 2 3

1 2 3

a a a

[a b c] b b b [ m n]

c c c

l

Perpendicular distance of a point with

p.v. a

from the plane through three points

whose position vectors are b, c,d

is

[bcd] [cad] [abd] [abc]

| b c c d d b |

The above relation also give the condition for

coplanarity of 4 points A, B, C & D with

position vectors a ,b, c,d

as

[a b c] [a c d] [b c d] [b d a]

Q. Prove distributive property of vector product:

a (b c) a b a c

.

Vector Triple Product :

(a b) c

is a vector which is coplanar with

a

and b

and perpendicular to c

.

(a b) c (a.c)b (b.c)a

Note : Unit vector coplanar with a

and b

and

perpendicular to c

is (a b) c

| (a b) c |

Q. 2

[a b b c c a] [abc]

a.a a.b a.c

b.a b.b b.c

c.a c.b c.c

Note that if a.b.c

are non coplanar vectors

then a b , b c

and c a

will also be non

coplanar vectors.

Q. If

1

2

3

V a (b c)

V b (c a) then prove that

V c (a b)

(i)1 2 3V , V , V

are coplanar

(ii)1 2 3V , V , V

from the sides of a triangle

(iii)1 2 3V V V

is a null vectors

(iv)1 2 3V , V , V

are linearly dependent.

Q. If 1ˆ ˆˆ â (b c) b2

where b and c are non

collinear then find the angle between a and

b , between a and c .

Q. If ˆ ˆ â i j k

, a·b 1

and ˆ â b j k

,

then find b

.



Q. Find a vector v

which is coplanar with the

vectors ˆ ˆ î j 2k & ˆ ˆ î 2 j k and is

orthogonal to the vector ˆ ˆ ˆ2 i j k It is

given that the projection of v

along the vector

ˆ ˆ î j k is equal to 6 3 .

Q. If a

and b

are vectors is space given by

ˆ î 2 ja

5

and ˆ ˆ ˆ2i j 3k

b14

, then find the

value of (2a b).[(a b) (a 2b)]

.

Q. Prove that :

ˆ ˆ ˆ ˆ ˆ î (a i) j (a j) k (a k) 2a

.

Q. If ˆ ˆ â 2i j 2k

, then the value of

2 2 2

ˆ ˆ ˆ ˆ ˆ î a i j a j k a k

is

equal to____ [JEE Main 2020]

Q. Given that q,p,b,a

are four vectors such

that 1)b(&0q.b,pba 2

, where

µ is a scalar then prove that

q.pa)q.p(p)q.a(

.

Q. If the vectors d,c,b

are not coplanar, then

prove that the vector

)cb()da()bd()ca()dc()ba(

is parallel to a

.

Q. Let x, y

and z

be three vectors each of

magnitude 2 and the angle between each

pair of them is 3

. If a

is a non-zero vector

perpendicular to x

and y z

and b

is a

non-zero vector perpendicular to y

and

z x

, then [IIT Advance 2014]

(A) b (b.z)(z x)

(B) a (a .y)(y z)

(C) a .b (a .y)(b.z)

(D) a (a .y)(z y)

12. Scalar Product of Four Vector :

a.c a.d

(a b).(c d) (a.c)(b.d) (a.d)(b.c)b.c b.d

Q. Prove that the acute angle between the two

plane faces of a regular tetrahedron is 1 1

cos3

.

Q. Prove that the angle between any edge and a

face of a regular tetrahedral is 1 1

cos3

Q. Prove that:{(a b) (a c)}.d (a.d)[a b c]

Q. Prove that : d.[a {b (c d)}] (b.d)[acd]

.

13. Vector Product of Four Vectors :

(a b) (c d)

= [a b d]c [a b c]d

[a c d]b [b c d]a

.

Note :

(i) (a b) (c d) 0

plane containing the

vectors a & b

and c

& d

are parallel.

Note : |||ly (a b).(c d) 0

the two planes are

perpendicular.

(ii) If a, b, c,d

are p.v. ‘s of four points then these

four points are in the same plane if

[a b d] [a b c] [a c d] [b c d]

.

Theorem in space :

If a, b, c

are 3 non zero non coplanar vectors

then any vector r

can be expressed as a linearcombination :

(i) r xa yb zc

(ii) r xa yb z(a b)

(iii) r x(b c) y(c a) z(a b)

Q. If a, b, c

, a b, b c

, c a

are non coplanar

vectors then prove that :

r.a r.b r.c

r (b c) (c a) (a b)[abc] [abc] [abc]

.



Q. Show that the p.v. of circumcentre of atetrahedron OABC is (where ‘O’ is the origin)

2 2 2a (b c) b (c a) c (a b)

2[abc]

.

Note :(i) Three non zero, non coplanar vectors are

linearly independent i.e. [abc] 0

.

Because if x 0, then xa yb zc 0

y za b c

x x

a

is coplanar with b

and

c

(contradiction)Hence x = 0, similarly y = 0 and z = 0.

(ii) Four or more vectors in 3D space are alwayslinearly dependent.

Q. a, b, c

are non-coplanar and d

is unit vector..

Find value of

a d b c b d c a c d a b

Q. Let b c c a a b

a ' ; b ' ; c '[a b c] [a b c] [a b c]

[a b c] 0

then show that :

(i) a.a ' b.b ' c.c ' 3

(ii) a a ' b b ' c c ' 0

Solving simultaneous vector equations for anunknown vectors :

There is no general method for solving suchequations, however dot or cross with known

or unknown vectors or dot with a b

,generally isolates the unknown vector. Use oflinear combination also proves to beadvantageous.

Q. Solve for x

, if x.a c

and a x b

, where

c is a non-zero scalar, a & b

are non-zerovectors.

Q. Given the vector a

and b

orthogonal to each

other find the vector v in terms of a

and b

satisfying v.a 0, v.b 1 and [v a b] 1

.

Q. Let a

and b

be two unit vectors such

that a b

= 0. For some x, y R, let

c xa yb (a b)

.

If | c |

= 2 and the vector c

is inclined at the

same angle to both a

and b

, then the valueof 8 cos2 is. [JEE Adv. 2018]

Q. Incident ray is along the unit vector v and the

reflected ray is along the unit vector w . The

normal is along unit vector a outwards.

Express w in terms of a and v .

[ JEE 2005 (Mains), 2 + 4 out of 60]

Q. Find the unknown vector R

satisfying

KR A R B

(K 0)

Q. Let x , y, z

be unit vectors such that

x y z a

, x (y z) b

,

(x y) z c

, 3

a.x2

, 7

a.y4

and

|a

| = 2. Find x, y, z

in terms of a, b, c

.

[REE – 96]

Q. If )0p(;b)ax(xp

prove that

xp b b a a p bxa

p p a

2

2 2

( . ) ( )

( ).

Q. Let a

, b

and c

be three vectors such that

a 3, b

= 5, b c

= 10 and the angle

between b

and c

is 3

. If a

is

perpendicular to the vector b c

, then

a b c

is equal to ______.

[JEE Main 2020]



THREE DIMENSIONALCOORDINATE GEOMETRY (3-D)

Distance / Section Formula :Any point P(x, y, z) then |x|, |y|, |z| areperpendicular distances from yz (x = 0),xz (y = 0), xy(z = 0) planes .Distance between 2 points

2 2 22 1 2 1 2 1(x x ) (y y ) (z z ) .

Co-ordinate of the point P which dividesAB in the ratio m : n is

2 1 2 1 2 1mx nx my ny mz nz, ,

m n m n m n

Direction Cosines :

Let 1 2 3

ˆ ˆ ˆ a = a i + a j + a k

be any vector which

makes angles with the +ve directionsOX, OY & OZ are called DIRECTIONANGLES & their cosines are called thedirection cosines.

D.C's are : 1 2a acos , m cos ,

| a | | a | l =

3an cos .

| a |

Note:

(i) ˆ ˆ â cos i cos j cos k

(ii) ˆ ˆ â i mj nk

(iii) ˆ ˆ ˆ ˆ ˆ â (a i)i (a j) j (a k)k

(iv) cos2 + cos2 + cos2 = 1

Direction ratios and its relationship withDirection Cosines :If a, b, c are 3 numbers which are proportionalto direction cosines are called the directionratio’s of a line i.e.

m n

a b c

l which gives

l = 2 2 2

a

a b c

; m =

2 2 2

b

a b c

;

n = 2 2 2

c

a b c

Note :(i) l, m, n gives a sense of direction.(ii) Direction ratios of a line joining two points

A and B are proportional tox2 – x1 , y2 – y1 ,z2 – z1.

(iii) Since

AB

= (x2 – x1) i + (y2 – y1) j + (z2 – z1) k

Hence the direction ratios of a vector

1 2 3ˆ ˆ â a i a j a k

are proportional to the

numbers a1, a2 and a3.If a line is having direction cosines l, m, n it

is travelling along the vector ˆ ˆ î mj nk l .

(iv) Angle between two lines with direction cosinesl1, m1, n1 and l2, m2, n2 is cos = l1l2 + m1 m2 + n1 n2

or cos = 1 2 1 2 1 2

2 2 2 2 2 21 2 3 1 2 3

a a b b c c

a a a b b b

(in terms of direction ratios)(a) If L1 is perpendicular to L2

then l1 l2 + m1 m2 + n1 n2 = 0or a1a2 + b1 b2 + c1 c2 = 0

(b) If L1 is parallel to L2 then 1 1 1

2 2 2

m n

m n

l

l

or 1 1 1

2 2 2

a b c

a b c

Q. Find the number of unit vectors which makesangle 30° with both x and y axis.

Q. Find the number of unit vectors which makesangles with x and y axis respectively,given that cos = 1/2, cos = 1/2

Q. A variable line has dc's l, m, n and l + l,m + m, n + n in two adjacent positions.If be the angle between the lines in thesetwo positions then prove that()2 = (l)2 + (m)2 + (n)2

Q. Find the direction cosines of a lineperpendicular to two lines whose dr’s are1, 2, 3 and –2, 1, 4

Q. The direction cosines l, m, n of two lines areconnected by the relations l + m + n = 0 and2lm + 2ln – mn = 0. Find them and the anglebetween them.



Q. A line makes angle with fourdiagonals of a cube. Prove that

cos2 + cos2 + cos2 + cos2 = 4

3.

PLANEA plane is a surface such that a line segmentjoining any two points on the surface lineswholly on it.

Different forms of the equations of plane :(i) Cartesian form :

A linear equation in three variables of the typeax + by + cz + d = 0 denotes the generalequation of a plane, where a, b, c are drs ofnormal to the plane.

Note :(i) Equation of xy, yz and zx planes are z = 0,

x = 0 and y = 0 respectively.(ii) The planes a1x + b1y + c1z + d1 = 0 and

a2x + b2y + c2z + d2 = 0 are

(a) Parallel if 1 1 1 1

2 2 2 2

a b c d

a b c d

(b) Perpendicular if a1a2 + b1b2 + c1c2 = 0

(c) Identical if 1 1 1 1

2 2 2 2

a b c d

a b c d

(ii) When a point and a vector normal to it isgiven

Hence 0(r r ) n 0

... (1)

where 0r

is the position vector of given point

and n

is normal vector to the plane. If

ˆ ˆ ˆr xi yj zk

and 0 1 1 1ˆ ˆ ˆr x i y j z k

then (1) becomesa(x – x1) + b (y – y1) + c (z – z1) = 0This is the equation of the plane containing the

point (x1 y1 z1) when ˆ ˆ ˆai bj ck is a vector

normal to it. where a, b, c are the dr’s of anormal to the plane.

(iii) Normal form of the plane

Let n be unit vector normal to the plane andperpendicular distance of the plane from theorigin is d.

ˆr.n d .....(i) (d > 0)

i.e. lx + my + nz = d (cartesian form)equation (i) helps us to know the distance ofthe plane from the origin and also the dc’s ofthe normal vector.

Q. ˆ ˆ ˆr.(6i 3j 2k) 1 0

(iv) Par ametr ic for m : r a p q

(v) Determinant form : 0r r p q 0

1 1 1

1 2 3

1 2 3

x x y y z z

p p p 0

q q q

(vi) Box form : 0r p q r p q

(vii) Intercept form of the plane :

x y z1

a b c where A (a, 0, 0) ; B (0, b, 0)

and (0, 0, c).

Note : Vector area

= 1 ˆ ˆ ˆ ˆ ˆ ˆab(i j) bc( j k) ca(k i)2

= 1 ˆ ˆ ˆbc i ca j abk2

i.e. Area of the ABC = 2 2 2 2 2 21

a b b c c a2

(viii) Equation of a plane through 3 non collinearpoints :Equation of a plane through 3 non collinear

points with p.v.’s a,b

and c

is

(r a).((b a) (c a)) 0

or [r a b a c a] 0

(r a).(a b b c c a) 0

or r.(a b b c c a) [a bc]

(ix) Equation of a plane through 3 points (xi, yi, zi)i = 1, 2, 3

1 1 1

2 1 2 1 2 1

3 1 3 1 3 1

x x y y z z

x x y y z z 0

x x y y z z

and conditions that 4 points (xi, yi, zi) i = 1, 2,3, 4 ; to be in the same plane is :

2 1 2 1 2 1

3 1 3 1 3 1

4 1 4 1 4 1

x x y y z z

x x y y z z 0

x x y y z z



Q. Find the equation of the plane throughthe point (2, –3, 1) and parallel to the plane3x – 4y + 2z = 5.

Q. The intercept made by the plane qn.r

on

the x-axis is

(A) n.i

q (B)

q

n.i

(C) qn.i

(D) |n|

q

Q. Find the equation of the plane which is parallelto the plane x + 5y – 4z + 5 = 0 and the sumof whose intercepts on the co-ordinate axesis 19 units.

Q. If from the point P (f, g, h) perpendiculars PL,PM be drawn to yz and zx planes then theequation to the plane OLM is

(A) 0h

z

g

y

f

x (B) 0

h

z

g

y

f

x

(C) 0h

z

g

y

f

x (D) 0

h

z

g

y

f

x

Q. The feet of normal from origin on a plane is and . Find the equation of the plane.

Q. Consider three vectors kjip

,

kj4i2q

and k3jir

. If

,p

q

and r

denotes the position vector of

three non-collinear points then the equationof the plane containing these points is(A) 2x – 3y + 1 = 0 (B) x – 3y + 2z = 0(C) 3x – y + z – 3 = 0 (D) 3x – y – 2 = 0

Q. Convert in Cartesian form

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆr i 2 j (2i j 3k) (3i 4 j k)

Perpendicular distance of a point P 0(r )

from a

plane r ×n = q

:

Distance = 0ˆ(r r ) n

0q r n

n

(Vector form)

Distance 1 1 1

2 2 2

ax by cz d

a b c

(Cartesian form){ Where equation of plane is ax + by + cz + d= 0 and P (x1, y1, z1) }

Distance between the parallel planes :

1 2

2 2 2

d d

a b c

where planes are ax + by + cz + d1 = 0 andax + by + cz + d2 = 0

Angle between two planes :The angle , between the two planes

1 1r.n q

and 2 2r.n q

, being equal to the

angle between the vectors 1n

and 2n

which

are normal to the planes, we have

1 1 2

1 2

n .ncos

| n || n |

1 1 2 1 2 1 2

2 2 2 2 2 21 2 3 1 2 3

a a b b c ccos

a a a b b b

.

Note: The angle , between the line r a b

and

1 b.nplane r.n q is sin

b n

.

Q. Find the equation of plane passing through(2, 2, 0) and (0, 3, 7) and parallel to y-axis.

Q. A tetrahedron has vertices P (1, 2, 1),

Q(2, 1, 3), R(1, 1, 2) and O(0, 0, 0). The

angle between the faces OPQ and PQR is :

[JEE Main 2019]

(A) 1 17

cos31

(B) 1 19

cos35

(C) 1 9

cos35

(D) 1 7

cos31

Q. Find the equation of plane passing through thepoint (1, 0, –2) and perpendicular to theplanes 2x + y – z = 2 and x – y – z = 3.



Q. If the plane 2x – 3y + 6z – 11 = 0 makes an

angle sin–1(k) with x-axis, then k is equal to

(A) 23 (B) 2/7

(C) 32 (D) 1

Q. A variable plane is at a constant distance p

from the origin and meets the coordinate axes

in points A, B and C respectively. Through

these points, planes are drawn parallel to the

coordinates planes. Find the locus of their point

of intersection.

Q. Find the equation of the plane parallel to

2x – 6y + 3z = 0 and at a distance of 2 from

the point (1, 2, –3).

Q. A plane which always remains at a constant

distance p from the origin cuts the co-ordinate

axes at A, B, C. Find the locus of

(a) Centroid of the plane face ABC

(b) Centre of the tetrahedron OABC

Two intersecting lines determine a unique plane:

Let the equations of the two lines are

r a b

r a c

where a

is position vector of

common point.

Hence the equation of the plane is,

(r a).(p c) 0

or [r a b c]

= 0 or

[r b c] [a b c]

.

Condition for coplanarity of two lines :

Let lines are r a b and r c d

( b

and d

are non collinear)

(r a).b d 0

is the equation of the plane

Condition is [a bd] [cbd]

.

Plane containing two parallel lines :

Let lines are r a c and r b c

(r a).{(a b) c} 0

or [r a c] [r c b] [a b c] 0

or [r a a b c] 0

.

Q. Find whether the two lines inersect or not.If they do, find the equation of the planecontaining them. If they don’t find theShortest Distacne between them.

(i) ˆ ˆ ˆ ˆ ˆr i j k (3i j)

ˆ ˆ ˆ ˆ& r 4i k (2i 3k)

(ii) ˆ ˆ ˆ ˆ ˆr 2 j k (i j k)

& ˆ ˆ ˆ ˆ ˆ ˆr 2i 3j 6k (2i j 5k)

(iii) ˆ ˆ ˆ ˆ ˆ ˆr i 2j 3k (2i 3j 2k)

& ˆ ˆ ˆ ˆ ˆ ˆr 2i 4j 5k (4i 6j 4k)

Plane containing a point B( b

) and a line

r = a + λp

, (point B not lying on the line):

(r a).{(a b) p} 0

Line of Intersection / Common Line between twointersecting planes:

ax b1y + c1z + d1= 0ax b2y + c2z + d2= 0then line isax b1y + c1z + d1 = ax b2y + c2z + d2= 0

Q. x + 2y – 3z = 0, 2x + y + z + 3 = 0

Q. If the lines x = ay + b, z = cy + d and

x = a'z + b', y = c'z + d' are perpendicular,

then : [JEE Main 2019]

(A) cc' + a + a' = 0

(B) aa' + c + c' = 0

(C) ab' + bc' + 1 = 0

(D) bb' + cc' + 1 = 0



Q. The shortest distance between the lines

x –1

0 =

y 1

–1

=

z

1 and x + y + z + 1 = 0,

2x – y + z + 3 = 0 is: [JEE Main 2020]

(A) 1 (B) 1

2

(C) 1

3(D)

1

2

Q. Let L1 and L

2 be the following straight

lines. 1 2

1 1 1 1: and :

1 1 3 3 1 1

x y z x y zL L and

1 2

1 1 1 1: and :

1 1 3 3 1 1

x y z x y zL L .Suppose the straight

line 1

:2

x y zL

l m

lies in the plane

containing L1 and L

2, and passes through the

point of intersection of L1 and L

2. If the line L

bisects the acute angle between the lines L1

and L2, then which of the following statements

is/are TRUE? [JEE Advanced 2020](A) (B) l + m = 2(C) (D) l + m = 0

Equation of the bisector planes between theplanes :

1 : a1 x + b1y + c1z + d1 = 0 and .....(i)2 : a2 x + b2y + c2z + d2 = 0 and .....(ii)

1 1 1 1 2 2 2 2

2 2 2 2 2 21 1 1 2 2 2

a x b y c z d a x b y c z d

a b c a b c

Acute/Obtuse angle bisectors can be easilyisolated by finding

cos = 1 2 1 2 1 2

2 2 2 2 2 21 1 1 2 2 2

a a b b c c

a b c a b c

,

where is the angle between any one of thetwo given planes and any one of the twobisector planes.

If 1

| cos | 12

is acute,

0 < cos < 1

2 is obtuse.

Vectorially :

Let 1 1r.n q

and 2 2r.n q

be the given

planes. Perpendicular distance of any point r

on either bisecting planes from the two givenplanes being equal, hence

1 2 1 2

1 2 1 2

n n q qr.

| n | | n | | n | | n |

where same sign is to be taken throughout.

Family of planes:The equation P1 + P2 = 0 gives the family ofplanes containing the line of intersection ofP1 = 0 and P2 = 0 for all R.

Q. Find the equation of plane through the line ofintersection of the planes x – 3y + 2z = 5 and2x – y + 3z – 1 = 0 which passing to point1, 2, 3.

Q. The plane x – y – z = 4 is rotated through 90o

about its line of intersection with the planex + y + 2z = 4. Find its equation in the newposition.

Q. Find the condition of lines x = ay + b,z = cy + d, x = a’y + b’, z = c’y + d’ areperpendicular.

Q. Find the reflection of the plane2x – 3y + 6z + 1 = 0 in the plane14x – 2y – 5z + 3 = 0.

Vectorially :To find the equation of the plane coaxal with

the planes 1 1r.n q

and 2 2r.n q

...(i)

and passing through the point with position

vector a

.

Q. Find the equation of plane through the

intersection of the planes ˆ ˆ ˆ ˆ ˆ ˆr (2i j 3k) 7 & r (2i 5j 3k) 9 0

and ˆ ˆ ˆ ˆ ˆ ˆr (2i j 3k) 7 & r (2i 5j 3k) 9 0

and origin.

Q. The plane lx + my = 0 is rotated about

its line of intersection with the plane z = 0

through an angle . Prove that the equation

to the plane in new position is

lx + my + tanmlz 22 = 0



Q. Find the equation of the plane containing the

line of intersection of the planes 1 1r.n q

,

2 2r.n q

and is parallel to the line of

intersection of the planes 3 3r.n q

and

4 4r.n q

Straight Line (in 3D)

Ist form r a b

IInd form 1 1 1x x y y z z

a b c

IIIrd form 1 1 1x x y y z z

m n

IVth form a

1x + b

1y + c

1z + d

1 = 0

= a2x + b

2y + c

2z + d

2

Parametric Form :x = x

1 + r cos ,

y = y1 + r cos ,

z = z1 + r cos

Note: Point on line at a distance r from (x1, y

1, z

1) is

given by (x1 + r l, y

1 + r m, z

1 + r n)

Some Common lines Perpendicular to Z axis :

1 1 1x x y y z z

cos cos cos / 2

1 1x x y y

cos cos

, z = z

1

Equation of X axis :

x 0 y 0 z 0

1 0 0

or y = 0 = z

any point on it (, 0, 0).

Equation of Y axis :

x = 0, z = 0 or x 0 y 0 z 0

0 1 0

Equation of line in two point form :

1 1 1

2 1 2 1 2 1

x x y y z z

x x y y z z

Q. Convert line into symmetrical form

r = (1, 2, 3) + (2, 3, 4). Also find the point

where line cut xz-plane.

Q.x 2 z 1

3 2

and y = 2, find :

(i) Angle with ‘y’ axis

(ii) Any trivial point on line

(iii) Distance from x z plane

(iv) Convert line into symmetrical form

(v) Point where line cut cordinate planes.

Q. If a point R(4, y, z) lies on the line segment

joining the points P(2, –3, 4) and Q(8, 0, 10),

then the distance of R from the origin is :

[JEE Main 2019]

(A) 2 14 (B) 6

(C) 53 (D) 2 21

Point of intersection of line in plane :

Line 1 1 1

1 1 1

x x y y z z

a b c

Plane ax + by + cz + d = 0

then point is a (x1 + a, y

1 + b, z

1 + c

where can be obtained.

Q. Find the point where the line

x 2 y 2 z 1

3 0 2

cuts the plane x + y + z = 0.

Q. Find equation of line passing through (0, 0, 0)

and to fixed line x 1 y 2 z

2 3 5

and

| | to fixed plane x – y + z + 2 = 0.



Q. Consider the following linear equations

ax + by + cz = 0

bx + cy + az = 0

cx + ay + bz = 0

Match the conditions/ expressions in Column

I with statements in Column II.

Column I Column II

(A) a + b + c 0 and (P) the equation

a2 + b2 + c2 = ab + bc + ca represent planes

meeting only at a

single point.

(B) a + b + c = 0 and (Q) the equation

a2 + b2 + c2 ab + bc + ca represent the line

x = y = z

(C) a + b + c 0 and (R) the equation

a2 + b2 + c2 ab + bc + ca represent identical

planes

(D) a + b + c = 0 and (S) the equation

a2 + b2 + c2 = ab + bc + ca represent the whole

of the three

dimensional space.

[JEE 2007, 3+3+3+3+6]

Distance of a point from line :

Q. Find the distance of P (1, 1, 1) from the line

x 1 y 1 z

2 1 1

.

Q. The vertices B and C of a ABC lie on the

line, x 2 y 1 z

3 0 4

such that BC = 5

units. The the area (in sq. units) of this triangle,given that the point A(1, –1, 2), is :

[JEE Main 2019]

(A) 2 34 (B) 34

(C) 6 (D) 5 17

Q. A line passing through the origin is

perpendicular to the lines [IIT-Advance 2013]

1 : (3 + t) i + (–1 + 2t) j + (4 + 2t) k , – < t <

2 : (3 + 2s) i + (3 + 2s) j + (2 + s) k , – < s <

Then, the coordinate(s) of the point(s) on 2

at a distance of 17 from the point of

intersection of and 1 is(are) :

(A) 7 7 5

, ,3 3 3

(B) (–1, –1, 0)

(C) (1, 1, 1) (D) 7 7 8

, ,9 9 9

Q. Consider a pyramid OPQRS located in the

first octant (x 0, y 0, z 0) with O as

origin, and OP and OR along the x-axis and

the y-axis, respectively. The base OPQR of

the pyramid is a square with OP = 3. The point

S is directly above the mid-point T of diagonal

OQ such that TS = 3. Then

[IIT Advance 2016]

(A) The acute angle between OQ and OS is

/3

(B) The equation of the plane containing the

triangle OQS is x – y = 0

(C) The length of the perpendicular from P to

the plane containing the triangle OQS is 3

2

(D) The perpendicular distance from O to the

straight line containing RS is 15

2

Q. Find equation of line passing through (1, 1, 2)

and | | to planes x + 2y + 2z + 3 = 0 and

x + 2y – z + 4 = 0.

Q. Find equation of line of intersection of planes

x + y – 2z = 8 and 3x – y + 4z = 12.



Q. Prove that the lines

3x + 2y + z – 5 = 0 = x + y – 2z – 3 and

2x – y – z = 0 = 7x + 10y – 8z are

perpendicular to each other.

Q. Find distance of point A(1, 0, –3) from

x – y – z = 9 measured parallel to line

x 2 y 2 z 6

2 3 6

.

Q. Find distance of the point A(3, 8, 2) from the

line x 1 y 3 z 2

2 4 3

measured parallel to

the : 3x + 2y –2z +15 = 0.

Q. Find equation of line passing through line

A(0, 1, 2) parallel to line

x 1 y 1 z 0

1 1 2

.

Q. The position vectors of the four angular

points of a tetrahedron OABC are (0, 0, 0);

(0, 0, 2); (0, 4, 0) and (6, 0, 0) respectively.

A point P inside the tetrahedron is at the same

distance 'r' from the four plane faces of the

tetrahedron. Find the value of 'r'.

Q. If the line 1 1 1

x y z

m n

, 2 2 2

x y z

m n

,

3 3 3

x y z

m n

are coplanar then prove that

1 1 1

2 2 2

3 3 3

m n

m n 0

m n

.

Q. Calculate minimum distance between two

skew lines x 1 y 3 z 3

2 3 4

and

x 2 y 4 z 5

3 4 5

.

Q. Find equation of line through (1, 1, 1) and

intersect the lines x 1 y 2 z 3

2 3 4

,

x 2 y 3 z 1

1 2 4

.

Q. Prove that the three planes 2x + y – 4z – 17 = 0,

3x + 2y – 2z – 25 = 0 and 2x – 4y + 3z + 25 = 0

intersect at a point and find the coordinate of

the points.

Q. Prove that three planes ˆ ˆ ˆr.(2i 5j 3k) 0

,

ˆ ˆ ˆr.(i j 4k) 2

& ˆ ˆr.(7 j 5k) 4 0

have a common line of intersection.

Foot of perpendicular and reflection of a point in

a plane :

Point (x1, y

1, z

1) and plane :

ax + by + cz +d = 0

Projection of line on a plane :

Image of line about plane :

Line of greatest slope :

It is a line in the given plane having maximum

angle with the horizontal plane.

Method :

For angle to maximum it must be to line of

intersection.

1 2Li N N

L

given = 1Li N

N1 N2



Q. Assuming the plane (4x – 3y + 7z) = 0 to be

horizontal plane, find the equation of line of

greatest slope in the plane 2x + y – 5z = 0 and

passing through the point (2, 1, 1).

Sphere :

(1) (x – u)2 + (y – v)2 + (z – w)2 = R2

Centre C (u, v, w) & radius R

Or r c R

General Equation :

x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0

represents a sphere here coefficient of

x2 = coefficient of y2 = coefficient of z2

and there is no terms of xy, yz and zx.

Its centre (–u, –v, –w),

radius = 2 2 2u v w d

Q. Find the sphere through (1, 0, 0), (0, 1, 0),

(0, 0, 1) having radius as small as possible.

Diametric form : (r a) (r b) 0

.

Q. Find the equation of sphere whose

one diametric ends ˆ ˆ ˆ(2i 6j 7k) and

ˆ ˆ ˆ(2i 4j 3k) .

Circle :

Intersection of sphere and plane.

Q. Find the radius of circular section in which the

sphere | r | 5

is cut by plane

ˆ ˆ ˆr.(i j k) 3 3

.

11. Vector & 3D_WB-2.pdf - Best Approach

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