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Vectors and the Three Dimensional Space

Chapter 1Lesson 1

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Learning Outcomes At the end of the lesson you should

be able to:1. Identify a vector in 3- space from

2-space and 1-space.2. Find the distance between points.3. Derive the equation of sphere.4. Define vector.5. Perform the arithmetic of vectors.6. Perform Dot and Cross product

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Rectangular Coordinate Systems

One way to identify points in a plane is to use a Cartesian coordinate system.

3- space: x , y, and z axes 2- space: x and y axes produces a Plane 1-space : x axes, a linez

y

xo

y

xo

xo

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OCTANTS

z

y

x ab

cP(a, b, c)

Z-axis : (0, 0, c)Y-axis : (0,b, 0 )X-axis : (a, 0, 0)

The coordinate axes define 3 coordinate plates: xy-plane, yz-plane and xz-plane.

The location of point P in space can be specified by listing 3 real numbers:

Mid-point The coordinate of the center between points A(a,b) and B(c,d) is given by

The coordinate of the point divides points A(a,b) and B(c,d) in a ratio m:n is given by

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2,2dbcaM

nmdmbn

nmcman ,

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Distance (d) in 3-space

22 ba

222 cbad

),,( 1111 zyxP

d c

ba

y

),,( 2222 zyxP

212

212

212 )()()( zzyyxxd

czzbyyaxx 121212 ;;

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SPHERE

222

2222 )0,0,0(,zyxr

Czyxr

z

y

x

rC(h,k,l)

P(x,y,z)

222 )()()( lzkyhxr

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Theorem: An equation of the form

where E, F, G and H are constant, represents a sphere , a point , or has no graph.

0222 HGzFyExzyx

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Example 1Find the center and radius of the sphere that has (1, -2, 4) and ( 3, 4, -12) as endpoints of a diameter.Answers: C (h,k,l) =( 2, 1, -4) d =17.3 ; r = 8.6

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Example 2Show that A(4, 5,2),B(1,7,3) and C(2,4,5) are vertices of an equilateral triangle.

AB=AC=BC= 14Answer:

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Simple Recall1. Find the distance between the

points A ( 2, 1, 3) and B (-1,4,1). 2. Find the equation of the sphere

with center at the origin and passing through a point (-1,-1, 2).

3. Find the center and radius of the sphere given

11246222 zyxzyx

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VECTORS Definition Vector is a quantity that has

both magnitude and direction usually represented by an arrow.

It has an initial point and a terminal point.Vector quantity: velocity,

displacement, force

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How to Draw a Vector? Example 1: v= <1, 3, 2> in 3-dimensional

Example 2: v= <4, -1> in 2-dimensional

Example 3 : a= 2i - 3j + 4k where i, j, k are unit vectors

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Equivalent Vectors Two vectors are equivalent if their corresponding components are equal.

Example: A= <2,-1,3> and B= < 2, -1, 3>

Vector A is equivalent to vector B !

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Sum of VectorsIf v and w are vectors , then the sum v + w is

v+w

vw

v+w = w+v0+v= v+0 =v

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Arithmetic Operations on Vectors Theorem. If

spaceinwvwvwvwvwwwwandvvvv

spaceforwvwvwvwwwandvvv

3,,,,,,

;2,,,

332211

321321

2211

2121

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If v is a nonzero vector and k is a nonzero real number (scalar) then the scalar multiple kv is defined to be a vector whose length is k times the length of v and whose direction is the same as that of v if k > 0 and opposite that of v if k< 0.

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spaceinwvwvwvwvspaceinwvwvwv

3,,2,332211

2211

If k is any scalar, then 32121 ,,, kvkvkvkvorkvkvkv

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Vectors with Initial Point Not the Origin Theorem.

space3in,,,ALSOspace2in,),(pointterminaland),(

pointinitialwithspace2invectoraisIf

12121221

121221

222111

21

zzyyxxPPyyxxPP

yxPyxPPP

lengththeis|| 21PP

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Norm of a Vector The norm or the magnitude (length) is denoted as

32123

22

21

2122

21

,,;space3in,;space2in

vvvvvvvv

vvvvvv

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Example 3:Find the norms of v=<3,-5>, 2vand w=<2, -4, 1>.Answers:

58.4;66.112;83.5 wvv

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Unit Vectors In 2-space: i =<1, 0 > ; j =<0 ,1> In 3-space: i=<1,0,0>; j=<0,1,0> ; k=<0, 0, 1>

i , j, k is known as the standard basis vector

Example: A= <-1,5,-6> = -i + 5j- 6z

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Normalizing a Vector vkvv

u 1

is a unit vector with the same direction as v.Example 4.Find the unit vector that has the same direction as v=2i-j+4kAnswer: 4,1,2

211u

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Vectors Determined by Angle

v

y

x

cosv

sinv

jvivvvv

sincossin,cos

v

Dot Product

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332211 vuvuvuvu

What is the dot product of u and v?

2,4,3say,,3,2,1say,,

321

321

vvvvvuuuuu

Note: the dot product of 2 vectors produces a scalar value

Cross Product

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21

21

31

31

32

32

321

321

321321 ,,&,,

vvuu

kvvuu

jvvuu

ivu

vvvuuukji

vu

vvvvuuuu

Note: The cross product of 2 vectors produces a vector!Cross Product is not commutative!

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Sine Law

sinsinsincba

cos2222 abbacCosine Law

c

ab