Find the vectorial equation of the line through $P$ and orthogonal to two planes

Question

I have to find the vectorial equation of the line through $P$ and orthogonal to $r:(x,y,z)=(1,−1,−1)+\lambda(1,−1,0)$ and $s:(x,y,z)=(\frac{3}{2},-\frac{1}{2},0)+\alpha(\frac{1}{2},\frac{1}{2},1)$.

Thank you very much!

Answer 1

Hint: one direction orthogonal to both lines is the vector ${\bf v} = (1,-1,1)\times\left(\frac{1}{2},\frac{1}{2},1\right)$. Then your line is ${\bf X}(t) = P+t{\bf v},$ with $t \in \Bbb R$.