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For the three planes given by bx - ay - n = 0, cy - bz = l, and az - cx = m, what condition must hold for them to have a common line of intersection?

1. a + b + c = 0
2. a = b = c
3. a + bm + cn = 0
4. l + m + n = 0

Correct answer: a + bm + cn = 0

Solution

Taking c*(bx-ay-n) + a*(cy-bz-l) + b*(az-cx-m), all the x, y, z coefficients cancel, leaving the consistency condition -(cn+al+bm)=0, i.e. al+bm+cn=0. This corresponds to the option containing bm+cn (al+bm+cn=0).

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