step 1

Objective: to find the intersection of the two lines and .

step 2

The projection (blue line ) of the red segment through perpendicular to onto has magnitude , where is the angle between the red segment and .

step 3

In vector terms, the tip of the blue vector is at a distance from , so that its position vector is , where is a unit vector (in purple). Use the positive sign whenever ; otherwise use the negative sign.

step 4

Let be the angle betwen the lines and . The point of intersection of the two lines (in orange) lies at the distance from (with ), so that its position vector is , where is a unit vector (in black). Choose the sign to give the position closest to .