Line perpendicular to given passing through point

[mitjak]

Line perpendicular to given line passing through a point

Homework Statement

What is the vector equation of a line passing through P(1,2,3) which is perpendicular to the line L1 : (x-4)/4 = (y-5)/5 = (z-6)/6

Homework Equations

Dot product
Projections
Planes, perhaps?

The Attempt at a Solution

I believe I am on the right track but what I'm interested in the other possible ways of tackling this.

Since we are given a point on the line in the equation of L1, let's call it Q(4,5,6), I thought I should attempt to project the vector PQ that connects the point on the line with the point P, onto the directional vector of the line. Then, adding the projection, QR, to the original directional vector I should be able to obtain the point on the line, R, where the vector PR is perpendicular to the line L1. From thereon, it's just a matter of using the point R and the vector PR to write out the line's equation.

Is my line of thinking correct, and what would be the alternative ways of solving this? The calculus teacher hinted at a number of ways, one of which would involve constructing a plane from the line L1, with the normal of the plane being the vector perpendicular to the line; and the other method having possibly to do with the dot product. I am not sure how one would go about solving this problem in those ways but I am very interested in finding out.

Advice much appreciated!

 

[mitjak]

I've used the dot product and Pythagorean theorem to confirm my answers, as well as Mac OS' Grapher to visualize them, and the answers seem correct, but I'm still curious how you would solve this using planes. Somehow planes seem more elegant. Call me a geek if you want but I'd like to understand this from a different angle :).