Question

Find the coordinates of the point where the line through (3, – 4, – 5) and (2, –3, 1) crosses the plane passing through three points (2, 2, 1), (3, 0, 1) and (4, –1, 0)

Answer 1

Equation of plane through three points (2, 2, 1), (3, 0, 1) and (4, –1, 0) is

`[(vec"r" - (2hat"i" + 2hat"j" + hat"k")]*[(hat"i" - 2hat"j") xx (hat"i" - hat"j" - hat"k")]` = 0

i.e. `vec"r"*(2hat"i" + hat"j" + hat"k")` = 7 or 2x + y + z – 7 = 0   ......(1)

Equation of line through (3, – 4, – 5) and (2, – 3, 1) is

`(x - 3)/(-1) = (y + 4)/1 = (z + 5)/6`  .....(2)

Any point on line (2) is `(-lambda + 3, lambda - 4, 6lambda - 5)`.

This point lies on plane (1).

Therefore,

Hence the required point is (1, – 2, 7).