Find the equation of the line passing through the point (4,-14,4) and intersecting the line of intersection of the planes : 3 x + 2 y − z = 5 and x − 2 y − 2 z = − 1 at right angles.

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TestHub · Accepted Answer

The intersection line has direction d. The proposed foot Q lies on both planes, and PQ is perpendicular to d, proving that it is the required intersection. d = ( 3 , 2 , − 1 ) × ( 1 , − 2 , − 2 ) = ( − 6 , 5 , − 8 ) . P = ( 4 , − 14 , 4 ) , Q = ( 7 , − 4 , 8 ) , 3 ( 7 ) + 2 ( − 4 ) − 8 = 5 , 7 − 2 ( − 4 ) − 2 ( 8 ) = − 1. Q − P = ( 3 , 10 , 4 ) , ( Q − P ) ⋅ d = − 18 + 50 − 32 = 0. r = ( 4 , − 14 , 4 ) r = + t ( 3 , 10 , 4 ) . ​

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