[Math] How to find parametric equation of the line which is perpendicular to 2 lines and passes through point of intersection

Question

$$
\left\{\begin{array}{lcccc}
\mbox{Line}\ 1 & : & x = 1 + 2a, & y = 2 – a, & z = 4 – 2a
\\[1mm]
\mbox{Line}\ 2 & : & \!\! x = 9 + b, & \,\,\, y = 5 + 3b,
& \,\,z = -4-b
\end{array}\right.
$$
Point of Intersection: $\left[7,-1,-2\right]$.

How to find parametric equation of the line which is perpendicular to these $2$ lines and passes though point of intersection ?.

Answer 1

Hint:

The coefficient vector of the parameter $a$ represents the "direction vector" of that line, and taking the cross product with another vector will give a perpendicular direction.