(i) We have;
l2 + m2 + n2 =1
cos2θ + cos2 θ + cos2 θ =1
= 3cos2 θ = 1
= cos2 θ = \(\frac{1}{3}\)
= cos θ = ± \(\frac{1}{\sqrt3}\)
Therefore direction cosines are
± \(\frac{1}{\sqrt3}\) ,± \(\frac{1}{\sqrt3}\) ,± \(\frac{1}{\sqrt3}\) ,
(b) Equation of plane is
(ii) Since the Plane x + py + qz = 0 is perpendicular to the Line \(\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{1}\) we have their direction ratios proportional. Plane dr’s is 1, p, q and dr’s of Line is 2, 3, 1.
⇒ \(\frac{1}{2}=\frac{p}{3}=\frac{q}{1}\\⇒p=\frac{3}{2},q=\frac{1}{2}\)