Q.Find the direction cosines of the line that is perpendicular to both the lines with direction ratios $1,-2,-2$ and $0,2,1$. Hence find the Cartesian equation of the line through the point $(2,1,-3)$ parallel to it.

A line perpendicular to both given lines must have direction ratios along $(1,-2,-2)\times(0,2,1)$:

so direction ratios $(2,-1,2)$, with magnitude $\sqrt{4+1+4}=3$. Direction cosines: $\left(\dfrac23,-\dfrac13,\dfrac23\right)$.

The line through $(2,1,-3)$ with these direction ratios: …

Use the cross product of the two given direction-ratio triples to obtain a direction perpendicular to both (the defining property of the cross product), normalise it for the direction co …

Sign errors in the cross-product cofactor expansion, particularly the middle ($\hat j$) term; writing $(z-3)$ instead of $(z+3 …