SOLUTION: The points A, B, C and D have position vectors i +j +k, 2i+3j, 3i+5j-2k and -j+k respectively. find which of the following pairs of lines are parallel. a) AB and CD b) BC and CD

Algebra ->  College  -> Linear Algebra -> SOLUTION: The points A, B, C and D have position vectors i +j +k, 2i+3j, 3i+5j-2k and -j+k respectively. find which of the following pairs of lines are parallel. a) AB and CD b) BC and CD       Log On


   



Question 1109182: The points A, B, C and D have position vectors i +j +k, 2i+3j, 3i+5j-2k and -j+k respectively. find which of the following pairs of lines are parallel. a) AB and CD b) BC and CD c) BC and AD
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Looking at the vector addition,
A%2BAB=B so
AB=B-A
Similarly,
BC=C-B
CD=D-C
AD=D-A
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A:(1,1,1)
B:(2,3,0)
C:(3,5,-2)
D:(0,-1,1)
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AB=B-A=(2,3,0)-(1,1,1)=(2-1,3-1,0-1)=(1,2,-1)
BC=C-B=(3,5,-2)-(2,3,0)=(3-2,5-3,-2-0)=(1,2,-2)
CD=D-C=(0,-1,1)-(3,5,-2)=(0-3,-1-5,1-(-2))=(-3,6,3)
AD=D-A=(0,-1,1)-(1,1,1)=(0-1,-1-1,1-1)=(-1,-2,0)
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You can either find the unit vectors of each vector and compare or you can look for vectors that are multiples of each other, Y=kX where k is an integer.
Upon inspection you can see that CD=-3%2AAB, so CD and AB are parallel.