If the lines \[\overrightarrow r = \left( {\widehat i - \widehat j + \widehat k} \right) + \lambda \left( {3\widehat j - \widehat k} \right)\] and \[\overrightarrow r = \left( {\alpha \widehat i - \widehat j} \right) + \mu \left( {2\widehat i - 3\widehat k} \right)\] are co-planar, then the distance of the plane containing these two lines from the point (\[\alpha\], 0, 0) is :
Solutions · 1
Official solution
\[\because\] Both lines are coplanar, so
\[\left| {α−102030−1−1−3 } \right| = 0\]
\[ \Rightarrow \alpha = {5 \over 3}\]
Equation of plane containing both lines
\[\left| {x−102y+130z−1−1−3 } \right| = 0\]
\[ \Rightarrow 9x + 2y + 6z = 13\]
So, distance of \[\left( {{5 \over 3},0,0} \right)\] from this plane
\[ = {2 \over {\sqrt {81 + 4 + 36} }} = {2 \over {11}}\]