Question

Show that the straight lines whose direction cosines are given by the equations al + bm + cn = 0 and ul² + vm² + wn² = 0 are perpendicular if a²(u + v) + b²(u + w) + c²(u + v) = 0 and parallel if (a²/u) + (b²/v) + (c²/w) = 0. 

Answer 1

We have, al + bm + cn = 0

(a) If two straight lines are parallel, the Eq. (i) will provide us equal roots.

Thus, D = 0

(b) When two straight lines are perpendicular, the, product of the roots = ((uc² + wa²)/(ub² + va²))  

Hence, the result.