I've some questions about the proof of
As you can see there are two parts to the proof.
Let be a line and let be a point. There is a unique point on for which is orthogonal to . Moreover, is given by
the following formulas:
where is any point on , is a unit vector in the direction of , and is a unit vector orthogonal to .
can be parameterized as (Vector equation of line). We seek a point for which is orthogonal to the direction of the line, i.e. for which
This relation has a unique solution Substituting for in gives the first
formula(first proof) for in the statement of the result.
My question is:
How did the author find the solution from the equation? What method he used for this?
How did he conclude that ?
And back to the proof.
gives the second part of the proof.
How to prove by substituting into ?