Question

Find the point of intersection of the lines AB and CD where A = (7, 6,1), B = (17,-18, -3),
C = (1, 4,-5) and D = (3, 4, 11).​

Answer 1

The given lines do not intersect each other.

Step-by-step explanation:

The given points are A (7, 6, 1), B (17, - 18, - 3), C (1, 4, - 5) and D (3, 4, 11)

The equations of the line AB are

(x - 7)/(17 - 7) = (y - 6)/(- 18 - 6) = (z - 1)/(- 3 - 1)

or, (x - 7)/10 = (y - 6)/(- 24) = (z - 1)/(- 4)

or, (x - 7)/5 = (y - 6)/(- 12) = (z - 1)/(- 2) = t (say)

Then x = 5t + 7, y = - 12t + 6, z = - 2t + 1 .....(1)

Again, the equations of the line CD are

(x - 1)/(3 - 1) = (y - 4)/(4 - 4) = (z + 5)/(11 + 5)

or, (x - 1)/2 = (y - 4)/0 = (z + 5)/16

or, (x - 1)/1 = (y - 4)/0 = (z + 5)/8 = u (say)

Then x = u + 1, y = 4, z = 8u - 5 .....(ii)

To find the point of intersection of the lines AB and CD, from (i) and (ii), we equate the corresponding values of x, y, z.

5t + 7 = u + 1 or, 5t - u + 6 = 0 .....(iii)

- 12t + 6 = 4 or, t = 1/6 .....(iv)

- 2t + 1 = 8u - 5 or, t + 4u - 3 = 0 .....(v)

Using (iv), from (iii), we get

5 (1/6) - u + 6 = 0

or, u = 5/6 + 6

or, u = 41/6

But the values t = 1/6 and u = 41/6 do not satisfy (v). So the lines do not intersect each other.

[ Refer to the image to see the lines not intersecting each other. ]

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