Q:

Vector lines that are perpendicular to each other

I have been given 2 lines 
L1 = < 3,- 1, 0> + x <2, 2 , -2>
L2 = < z, 0 , -2> + y <-1, B , 1>
I was asked to find z and B given that both lines intersect and they are perpendicular to eachother.

A:

Perpendicular, so the direction vectors should have a dot product equal to $0$:
$$(2,2,-2) \cdot (-1,B,1) = 0  \iff \color{blue}{B =\ldots}$$
Then with that value of $B$, look for a point where the lines meet:
$$(3,-1,0)+x(2,2,-2) = (z,0,-2)+y(-1,\color{blue}{B},1) \iff \left\{\begin{array}{rcl}
3+2x = z-y \\
-1+2x = \color{blue}{B}y \\
-2x = -2+y
\end{array}\right. \iff \cdots$$
This is a system of three equations in the unknowns $x$, $y$ and $z$.

