C1L1 Vectors and the Three Dimensional Space

8/11/2019 C1L1 Vectors and the Three Dimensional Space

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Vectors and the ThreeDimensional Space

Chapter 1

Lesson 1


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Learning Outcomes

At the end of the lesson you should be ableto:

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 zaxes 2- space: xand yaxes produces a Plane

1-space : xaxes, a line

z

y

x

o

y

xo

xo


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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,

2

dbcaM

nm

dmbn

nm

cman,


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Distance (d) in 3-space

22 ba

222 cbad

),,( 1111 zyxP

d c

b

a

y

),,( 2222 zyxP

2

12

2

12

2

12 )()()( zzyyxxd

czzbyyaxx 121212 ;;


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SPHERE

222

2222 )0,0,0(,

zyxr

Czyxr

z

y

x

r

C(h,k,l)

P(x,y,z)

222 )()()( lzkyhxr


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Theorem: An equation of the form

where E, F, Gand Hare constant,represents a sphere , a point , or has nograph.

0222 HGzFyExzyx
http://localhost/var/www/apps/conversion/tmp/scratch_10/C1L1%20Vectors%20and%20the%20Three%20Dimensional%20Space%20-%20Theorem.mwshttp://localhost/var/www/apps/conversion/tmp/scratch_10/C1L1%20Vectors%20and%20the%20Three%20Dimensional%20Space%20-%20Theorem.mws


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Example 1

Find the center and radius of thesphere that has (1, -2, 4) and

( 3, 4, -12) as endpoints of adiameter.

Answers: C (h,k,l) =( 2, 1, -4)d =17.3 ; r = 8.6


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Example 2

Show that A(4, 5,2),B(1,7,3)and C(2,4,5) are vertices of

an equilateral triangle.

AB=AC=BC= 14

Answer:


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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 withcenter at the origin and passing through apoint (-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 bothmagnitude 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= in 3-

dimensional

Example 2: v= in 2-dimensional

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

kare unit vectors


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Equivalent Vectors Two vectors are equivalent if theircorresponding components are equal.

Example:

A= 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

v

wv+w = w+v

0+v= v+0 =v


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Arithmetic Operations on

Vectors Theorem. If

spaceinwvwvwvwv

wwwwandvvvv

spaceforwvwvwv

wwwandvvv

3,,

,,,,

;2,

,,

332211

321321

2211

2121


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If v is a nonzero vector and k is anonzero real number (scalar) then thescalar multiple kv is defined to be avector whose length is k times thelength of v and whose direction is the

same as that of v if k > 0 and oppositethat of v if k< 0.


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spaceinwvwvwvwv

spaceinwvwvwv

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,,,ALSO

space2in,

),(pointterminaland),(

pointinitialwithspace2invectoraisIf

12121221

121221

222111

21

zzyyxxPP

yyxxPP

yxPyxP

PP

lengththeis|| 21PP


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Norm of a Vector Thenorm or the magnitude (length) is

denoted as

321

2

3

2

2

2

1

21

2

2

2

1

,,;space3in

,;space2in

vvvvvvvv

vvvvvv


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Example 3:

Find the norms of v=, 2v

and w=.

Answers:

58.4;66.112;83.5 wvv


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Unit Vectors In 2-space:

i = ; j =

In 3-space:i=; j= ; k=

i , j, k is known as the standardbasis vector

Example: A= = -i + 5j- 6z


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Normalizing a Vectorvkv

vu

1

is a unit vector with the same direction as v.

Example 4.Find the unit vector that has the samedirection as v=2i-j+4k

Answer: 4,1,221

1u


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Vectors Determined by

Anglev

y

x

cosv

sinv

jvivv

vv

sincos

sin,cos

v


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Sine Law

sinsinsin

cba

cos2222 abbac

Cosine Law

c

a

b

C1L1 Vectors and the Three Dimensional Space

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