shortest diatance between plane and tower

Here's a calc III problem(that was my approach) if anyone would like a shot. Not particularly difficult but kind of interesting.

"An airplane approaches the end of a runway from the east with an angle of 20 degrees. A tower 50 feet high is located 400 feet south and 800 feet east of the runway. How close does the plane come to the top of the tower?".

Here's a 'paint' drawing. It's a little tilted, but it should suffice.

No one wanted to tackle the problem, huh?.

Well, here's is the way I done it. I was wondering if someone else had a different method or verification of my solution.

We want the distance of the point from the line of descent of the plane. The point is on the line and the line has direction vector

Then we have





The distance of P1 from the line using the distance from a point to a line formula.