prove that the lines whose direction cosines are given by

ul^2+vm^2 wn^2=0 and al+bm+cn=0 are

(i) perpendicular if u(b^2+c^2)+v(C^2+a^2)+w(a^2+b^2)=0

(ii) parallel if a^2/u + b^2/v + c^2/w =0.

Dear student,

u1 = [l1, m1, n1] is a unit vector along one line
u2 = [l2, m2, n2] is a unit vector along the other line

The cross product, u1 x u2 is perpendicular to both and is a unit vector. Its components are

[m1n2 − m2n1, n1l2 − n2l1, l1m2 ­− l2m1]