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Question

Using vectors show that the point A (2,3,5), B (7,0,1) C (3,2,5) and D (3,4,7) are such that AB and CD intersect at the point P(1,2,3).

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Solution

Given points are
A(-2 , 3 , 5) B(7 , 0 , 1) C(-3 , -2 , -5) D(3 , 4 , 7)

line passing through the points A and B is

¯¯¯¯A+t(¯¯¯¯B¯¯¯¯A)2ˆi+3ˆj+5ˆk+t(9ˆi3ˆj4ˆk)(9t2)ˆi+(33t)ˆj+(54t)ˆk(1)linepassinthrough pointCandD¯¯¯¯C+S(¯¯¯¯¯D¯¯¯¯C)3ˆi2ˆj5ˆk+S(6ˆi+6ˆj+12ˆk)(6S3)ˆi+(6S2)ˆj+(12S5)ˆk(2)

given that equation (1) and (2) intersect each other at a point , so by equating the term , we get

9t2=6S3ˆicoefficent33t=6S2ˆjcoefficent12t5=112t=4t=13puttvalueinˆkcoefficent5413=12S5263=12SS=1318

So , equation og the line will be

(9×132)ˆi+(3313)ˆj+(54×13)ˆkˆi+2ˆj+113ˆkandalso43ˆi+73ˆj+113ˆk

point of intersection (1 , 2 , 3)



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