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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)
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Solution
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).
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