Collinearity

As with 2D - line segments are collinear if they share a point and are multiples of the same vector.

Ex Prove that the points F(-7,1,3), G(-3,-2,10) and H(9,-11,31) are collinear and find the ratio of FG:GH.

*********

FG = g – f = ( ) - ( )-3-2

-7 1 = ( )

4 -3

GH = h – g = ( ) - ( ) 9-11

-3-2 = ( )12

-9 = 3( ) 4

-3

FG & GH are multiples of the same vector and have a common point G so the three points are collinear.

10 3 7

31 10 21 7

FG

H

1

3

FG:GH = 1:3

also FG:FH = 1:4

GH:FH = 3:4

FH:HG = 4:-3

GF:GH = -1:3 etc

NB: these ratios take direction into account so include negatives

The Section Formula

Reminder

Ex A is (2,-6,1). Find B if AB = ( ). 2

-3-4

**********

B is A + AB = (2,-6,1) + ( ) 2

-3-4

= (4,-9,-3)

Add x-coord & x-component, y-coord & y-component, z-coord & z-component.

Ex NABThe point T divides PQ in the ratio 3:1. Find T when P is (-3,5,1) & Q is (13,-3,25).

*********

P

TQ

3

1

PT = 3/4PQ = 3/4(q – p) = 3/4{( ) - ( )}13-325

-3 5 1

= 3/4( )16-824

= ( )12-618

T is P + PT = (-3,5,1) + ( )12-618 = (9,-1,19)