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Chapter 1: Vectors and the

Geometry of Space

Section 1.2Space Coordinates and Vectors in Space

Dr. Maslan bin Osman

Mathematics Department, Faclty of Science,!"M

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#n this lesson yo $ill learn:

o% Space & "he three&dimensional coordinate systemo 'oints in space, ordered tripleso"he distance bet$een t$o points in spaceo"he midpoint bet$een t$o points in spaceo"he standard form for the e(ation of a sphere

oVectors in % SpaceoDifferent forms of )ectorsoVector operationso'arallel )ectorso*pplications of )ectors

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're)iosly yo stdied )ectors in the Cartesian plane or 2&dimensions, no$$e are +oin+ to epand or -no$led+e of )ectors to %&dimensions. efore$e discss )ectors, let/s loo- at %&dimensional space.

"o constrct a %&dimensional system, start $ith a y0 plane flat on the paperor screen.

y

0

3et, the &ais is perpendiclar

thro+h the ori+in. "hin- of the&ais as comin+ ot of the screento$ards yo.

For each ais dra$n the arro$represents the positi)e end.

"hree&Dimensional Space

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y

0

"his is considered a ri+ht&handed system.

"o reco+ni0e a ri+ht&handed system, ima+ine

yor ri+ht thmb pointin+ p the positi)e 0&ais,yor fin+ers crl from the positi)e &ais to thepositi)e y&ais.

#n a left&handed system, if yor left thmb is pointin+ p the positi)e 0&ais,yor fin+ers $ill still crl from the positi)e &ais to the positi)e y&ais. elo$is an eample of a left&handed system.

0

y

"hro+hot this lesson, $e $ill seri+ht&handed systems.

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y

0

"he %&dimensional coordinate system is di)ided into ei+ht octants. "hreeplanes sho$n belo$ separate % space into the ei+ht octants.

"he three planes are the y0 plane $hich isperpendiclar to the &ais, the y plane $hichis perpendiclar to the 0&ais and the 0 plane$hich is perpendiclar to the y&ais.

"hin- abot 4 octants sittin+ on top of the y

plane and the other 4 octants sittin+ belo$ they plane.y0 plane

y

0

y plane

y

0

0 plane

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5)ery position or point in %&dimensional space is identified by an ordered

triple,, y, 0.

6ere is one eample of plottin+ points in %&dimensional space:

'lottin+ 'oints in Space

y

0

' %, 4, 2

"he point is % nits in front of the y0 plane,4 points in front of the 0 plane and 2 nitsp from the y plane.

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6ere is another eample of plottin+ points in space. #n plottin+ the point 7&%,4,&8 yo $ill need to +o bac- from the y0 plane % nits, ot from the 0plane 4 nits and do$n from the y plane 8 nits.

y

0

7 &%, 4, &8

*s yo can see it is more difficlt to )isali0e points in % dimensions.

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Distance et$een "$o 'oints in Space

"he distance bet$een t$o pointsin space is +i)en by the formla:

( ) ( ) ( )212

2

12

2

12 00yy,,d ++=

( ) ( )222111

,,and,,0y,70y,'

"a-e a loo- at the net t$o slides to see ho$ $e come p $ith this formla.

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Consider findin+ the distance bet$een the t$o points,.

#t is helpfl to thin- of a rectan+lar solid $ith ' in the bottom bac-corner and 7 in the pper front corner $ith 9 belo$ it at .

( ) ( )222111 ,,and,, 0y,70y,'

'

7

9

!sin+ t$o letters to represent the

distance bet$een the points, $e -no$from the 'ytha+orean "heorem that'7 ; '9< 97

!sin+ the 'ytha+orean "heorem a+ain

$e can sho$ that

'9 ;

( )122 ,, 0y,

( ) ( )212

2

12 yy,, +

( )12

,,

( )12

yy

3ote that 97is .( )12 00

( )12

00

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'

7

9

( )12

,,

( )12

yy

( )12

00

Startin+ $ith '7 ; '9< 97

Ma-e the sbstittions: '9 ; and 97;( ) ( )2

12

2

12 yy,, + ( )12 00

"hs, '7 ;

Or the distance from ' to 7,

'7;

( ) ( ) ( )212

2

12

2

1200yy,, ++

( ) ( ) ( )212

2

12

2

1200yy,, ++

"hat/s ho$ $e +et the formla for

the distance bet$een any t$o pointsin space.

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Find the distance bet$een the points '2, %, 1 and 7&%,4,2.

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

2.8%%2=

1128

118

12%42%

222

222

2

12

2

12

2

12

==

++=

++=

++=

++=

d

d

d

d

00yy,,d

5ample 1:

>e $ill loo- at eample problems related to the three&dimensionalcoordinate system as $e loo- at the different topics.

Soltion: 'l++in+ into the distance formla:

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5ample 2:

Find the len+ths of the sides of trian+le $ith )ertices ?, ?, ?, 8, 4, 1 and4, &2, %. "hen determine if the trian+le is a ri+ht trian+le, an isoscelestrian+le or neither.

Soltion: First find the len+th of each side of the trian+le by findin+ thedistance bet$een each pair of )ertices.

?, ?, ? and 8, 4, 1

( ) ( ) ( )

42

11@28

?1?4?8 222

=

++=

++=

d

d

d

?, ?, ? and 4, &2, %

( ) ( ) ( )

2A

A41@

?%?2?4 222

=

++=

++=

d

d

d

8, 4, 1 and 4, &2, %

( ) ( ) ( )

41

4%@1

1%4284 222

=

++=

++=

d

d

d

"hese are the len+ths of the sides of the trian+le. Since none of them aree(al $e -no$ that it is not an isosceles trian+le and since$e -no$ it is not a ri+ht trian+le. "hs it is neither.

( ) ( ) ( )222 412A42 +

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"he Midpoint et$een "$o 'oints in Space

"he midpoint bet$een t$o points, is +i)en by:( ) ( )222111 ,,and,, 0y,70y,'

+++=

2,

2,

2Midpoint 212121

00yy,,

5ach coordinate in the midpoint is simply the a)era+e of the coordinatesin ' and 7.

=

=

+++

1,2

=,1

2

2,

2

=,

2

2

2

2?,

2

4%,

2

42:Soltion

5ample %: Find the midpoint of the points '2, %, ? and 7&4,4,2.

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5(ation of a Sphere

* sphere is the collection of all points e(al distance from a center point.

"o come p $ith the e(ation of a sphere, -eep in mind that the distance

from any point , y, 0 on the sphere to the center of the sphere,

is the constant r $hich is the radis of the sphere.

!sin+ the t$o points , y, 0, and r, the radis in the distance

formla, $e +et:

( )ooo 0y, ,,

( ) ( ) ( )222r ooo 00yy,, ++=

#f $e s(are both sides of this e(ation $e +et:

"he standard e(ation of a sphereis

$here r is the radis and is the center.

( ) ( ) ( )2222r ooo 00yy,, ++=

( )ooo 0y, ,,

( )ooo 0y, ,,

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5ample 4:

Find the e(ation of the sphere $ith radis, r ; 8 and center, 2, &%, 1.

Soltion: Bst pl++in+ into the standard e(ation of a sphere $e +et:

( ) ( ) ( ) 281%2 222 =+++ 0y,

5ample 8:

Find the e(ation of the sphere $ith endpoints of a diameter 4, %, 1 and&2, 8, =.

Soltion: !sin+ the midpoint formla $e can find the center and sin+ thedistance formla $e can find the radis.

( )4,4,1

2=1,

28%,

224Center

=

+++= ( ) ( ) ( )

1A

A1A414%149adis

222

=

++=++=

"hs the e(ation is: ( ) ( ) ( ) 1A441 222 =++ 0y,

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5ample @:

Find the center and radis of the sphere, .?=C@4222 =++++ 0y,0y,

Soltion: "o find the center and the radis $e simply need to $rite thee(ation of the sphere in standard form, ."hen $e can easily identify the center, and the radis, r. "o dothis $e $ill need to complete the s(are on each )ariable.

( )ooo 0y, ,,

( ) ( ) ( )2222r ooo 00yy,, ++=

( ) ( ) ( ) %@4%21@A4=1@CA@44

=C@4

?=C@4

222

222

222

222

=++++

+++=+++++++

=++++

=++++

0y,00yy,,

00yy,,

0y,0y,

"hs the center is 2, &%, &4 and the radis is @.

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Vectors in "hree&Dimensional Space

3o$ that $e ha)e an nderstandin+ of the three&dimensional system, $eare ready to discss )ectors in the three&dimensional system. *ll theinformation yo learned abot )ectors in the pre)ios lesson $ill apply,only no$ $e $ill add in the third component.

Vectors in component form in three dimensions are $ritten as orderedtriples, in other $ords, no$ a )ector in component form is .>

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Gi)en the initial point, and the terminal point, , thecomponent form of the )ector can be fond the same $ay it $as on theCartesian 'lane.

( )%21

,, ppp' ( )%21

,, (((7

>

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More on Vectors in "hree&Dimensions

et and let c be a scalar.>+++

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et/s loo- at some eample problems in)ol)in+ )ectors.

5ample 1:S-etch the )ector $ith initial point '2, 1, ? and terminal point 7%, 8, 4."hen find the component form of the )ector, the standard nit )ector formand a nit )ector in the same direction.

Soltion: First dra$ a %D system and plot ' and 7. "he )ector connects ' to 7.

'

7

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>

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5ample 2:Gi)en the )ectorsfind the follo$in+:

a. b. c.

>++++++