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Collinearity

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. *********. ( ) - ( ). -3. -7. = ( ). 4. FG =.

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Collinearity

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  1. 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. ********* ( ) - ( ) -3 -7 = ( ) 4 FG = g – f = -2 1 -3 10 3 7 ( ) - ( ) = ( ) = 3( ) 9 -3 12 4 h – g = GH = -11 -2 -9 -3 31 10 21 7 FG & GH are multiples of the same vector and have a common point G so the three points are collinear.

  2. H 3 G 1 F FG:GH = 1:3 also FG:FH = 1:4 NB: these ratios take direction into account so include negatives GH:FH = 3:4 FH:HG = 4:-3 GF:GH = -1:3 etc

  3. The Section Formula Reminder Ex A is (2,-6,1). Find B if AB = ( ). 2 -3 -4 ********** (2,-6,1) + ( ) 2 B is A + AB = = (4,-9,-3) -3 -4 Add x-coord & x-component, y-coord & y-component, z-coord & z-component.

  4. Ex The point T divides PQ in the ratio 3:1. Find T when P is (-3,5,1) & Q is (13,-3,25). NAB ********* Q 1 T 3 P = 3/4{( ) - ( )} = 3/4( ) = ( ) 13 -3 16 12 PT = 3/4PQ = 3/4(q – p) -3 5 -8 -6 25 1 24 18 (-3,5,1) + ( ) 12 T is P + PT = -6 = (9,-1,19) 18

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