STPM Mathematics T / A Level - Vectors - WordPress.com

STPMMathematicsT / A Level

M.K.Lim

STPM Mathematics T / A LevelVectors

M.K.Lim

August 19, 2012

M.K.Lim STPM Mathematics T / A Level


M.K.Lim

Representation of Vectors

B

A

Definition

A vector is a quantity which has magnitude and specificdirection in space.

A quantity with magnitude but no direction is called a scalar.Examples of vectors are displacement,velocity and acceleration.We use ~AB is known as the displacement from A to B.Displacement is move from A to B as shown Vector ~AB can becalled vector a. Note the arrowhead points(direction) from Atowards B.



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Equivalent Displacements Contd

CA

B

+ sign means ’together with’= sign means ’is equivalent to’Vector equation ~AB + ~BC = ~ACWe can read as ’AB together with BC is equiv to AC’



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Negative vectors

Definition

If two vectors a and b ,have the same magnitude but oppositedirections a = −b .We say vectors a and b are equal lengthand opposite direction.

a

b



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Modulus of a Vector

Definition

The modulus of a vector is its magnitude.It is written as |a|. This is the length of the line represented.

Modulus or magnitude example

If a = 3i + 4j + 5kthen |a| =

√32 + 42 + 52 =

√50 = 5

√2



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Scalar Multiplication of a Vector

Definition

If λ is positive real number , then λ is a vector in the samedirection as a and of magnitude λa.It is natural that −λa is in opposite direction.Example ~PQ = 2a has same direction as a but twice itsmagnitude than ~AB.

Aa

B

Q

P We can say in this case λ is 2.



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Addition of Vectors- Triangle Law

CB

A

p

q

p + q

Definition

If vector p and vector q are two vectors, then the ResultantVector is p + q as represented by side AC. This is the vectorlaw of addition.

Note that the arrow point towards C for the resultant p + q.



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Addition Law Triangle Law Contd...

Its the head-to-tail story...~AB + ~BC = ~AC

If side AB represents vector p

Side BC represented by vector q

Then side AC is the resultant, as p + q going from tail ofp to head of q.

Note :The tail of vector q follows the head of vector p



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Addition Law Using Parallelogram ABCD

B

D

C

A

a a

b

b

Definition

Parallel sides AB and DC represented by vector aSimilarly, parallel sides BC and AD represented by vector bIn the triangle ABC, resultant ~AC = a + bIn triangle ADC , ~AC = a+ bTherefore a + b = b + aSince AC is the common between 4 ABC and 4 ACD



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Vectors Illustrated in Cartesian coordinates

i

j

0 1 2 3 4

1

2

3

4

5

A

C

Vector a is ~OA = 3i + 4j ,vector b is ~OC = i + 2jResultant vector :Aligning head of vector a with the tail ofvector b, so it becomes a + b = a + b hence ~OA + ~OC = ~CA



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Area of Parallelogram Using Vector Product

i

j

1 2 3 4

1

2

3

4

5

6

B

CD

A

h

~AB

~AD

θ



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Angle between two vectors

Angle between two vectors is unique labelled as θ.

Two vectors a and b are shown with angle in between.

It is the angle between the directions when the both linesconverge or diverge from a point shown as a blue dot. It isonly angle θ and not any other.

a

θb



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Unit Vector

Definition

Given a is a vector .The unit vector is written as a.A unit vector is a vector whose length is 1, so magnitude of ais 1.Therefore a =

a

|a|

A unit vector is in the direction of v is vector over itsmagnitude.Applied to Cartesian Coordinates, i is the unit vector in Oxdirection and j is the unit vector in Oy direction.



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Scalar or Dot Product

Definition

The scalar product of two vectors a and b is defined as ab cos θwhere θ is the angle between thema.b = ab cos θSometimes it is also known as Dot Product



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Vector or Cross Product

Definition

The vector product of two vectors a and b is defined as ab sin θwhere θ is the angle between them|a× b| = ab sin θSometimes the cector product is also known as the CrossProductThis product acts in a direction perpendicular to both a and b



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Vector Product of two vectors a and b

If two vectors are parallel,then θ = 0◦,then |a× b| = 0

If two vectors are perpendicular,then θ=90◦,then |a× b| =ab since sin 90◦ = 1



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Parallel Vectors

a

b

π

Definition

Two vectors a and b are parallel, then ab = ab cosπThen a.b = - a.b since cos 180◦ = −1For unit vectors,i.i = j.j = k.k = 1



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Perpendicular Vectors

Definition

Two vectors a and b are perpendicular, then ab = abcos π2 = 0

Then a.b = 0 since cos 90◦ = 0For unit vectors i.j = j.k = k.i = 0



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Cartesian Unit Vectors

Definition

Now i is the unit vector in direction of OxNow j is the unit vector in direction of OyNow k is the unit vector in direction of Oz



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Equation of a Line

In terms of two types

Vector Equation of a line

Cartesian Equation of a line



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Vector Equation of a Line

We want to get a vector equation of the blue line shownlater

This line is parallel to a direction vector b which shows thedirection

Recall the straight line equation y = mx + c

Similarly, we can use vectors to find the equation of a line

Consider a line parallel to vector b which passes through afixed point A with position vector a

Vector b is the direction vector for the line

We shall see the development of r = a + λb



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Vector Equation of a Line Contd...

If r is the position vector ~OP then ~AP = λb

where λ is a scalar parameter.

Now ~OP = ~OA + ~AP

Therefore we have r = a + λb

This equation gives the position of one point on the line

That is P is on the line ⇔ r = a + λb

r = a + λb



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Vector Equation of a Line

A

P(x , y , z)

b

O

r

a

x

y



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Equations of a Plane

Two types namely, Vector and Cartesian

Plane (green) is made reference to the origin.



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Vector Equation of a Plane

Definition

Plane ( green ) is defined as distance d from origin O and isperpendicular to unit vector n shown.

N

P

O

d

n

r

x



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Standard form of Vector equation of a Plane

If ON is perpendicular to the plane then, for any point P on theplane , NP is perpendicular to ON.If r is position vector of P,then ~ON = dn.Since P is on theplane,it means that ~NP. ~ON = 0The equation is called the scalar product form of the vectorequation of a plane.If r is a position vector,then ~NP =r−dn.Therefore it becomes (r - dn) . dn = 0This implies that r.n− dn.n = 0But n.n = 1,So that means

r.n = d

The equation is the standard form of a vector of a plane.



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Cartesian Equation of Plane

Consider a plane whose vector equation of a plane is r.n = d,where n = l i + mj + nk ,Now if a point P (x,y) is on the plane its position vectorr = xi + yj + zk satisfies the equation, so(x i + y j + zk).(l i + mj + nk = d ⇒ lx + my + nz = dThis is the Cartesian Equation of a Plane

lx + my + nz = d



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Summary -Vector Cartesian equations for Linesand Plane

Line:

r = a + λb

x − a1b1

=y − a2b2

=z − a3b3

Plane:

lx + my + nz = d


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