Find the equation for the line that satisfies the following:

Question

1. being parallel to the plane $P:x+2y-3z=1$
2. intersects orthogonally with the line $k:(x,y,z)=(1+2t,t,-1)$,$t\in R$.
3. intersects with the x-axis in any point.

I must be missing out on some information, because I end up with two unknowns when there should be none according to the correct answer.

Any help is greatly appreciated.

Answer 1

The only vectors which are normal to (1,2,−3) and (2,1,0) are multiples of (1,2,−3)×(2,1,0)=(3,−6,−3). Take v=(1,−2,−1) and a point in the x axes (λ,0,0) such that the line (x,y,z)=sv+(λ,0,0) intersects the line k.

I'm glad it helped you. The cross product is very useful for 3D geometry problems.