Proof of w = u + {(p-u).T}T = p - {(p-u).N}N and couple of questions?

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 .

Proof:

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.

Substituting into:

gives the second part of the proof.

How to prove by substituting into ?

Re: Proof of w = u + {(p-u).T}T = p - {(p-u).N}N and couple of questions?

Don't worry I've found the answers.

For the first question i had: we know that is a unit vector. So

If I plugin into I get:

Now for the second part of the proof if I plugin into I get:

And that's it. Q.E.D.

Re: Proof of w = u + {(p-u).T}T = p - {(p-u).N}N and couple of questions?

I understand the proof. But what makes one think that is actually the projection of which is and not just any other point on the line ?

Because why is this not a projection of point ? Why do only that satisfies this have to be the projection of ?

Re: Proof of w = u + {(p-u).T}T = p - {(p-u).N}N and couple of questions?

Quote:

Originally Posted by

**x3bnm** I understand the proof. But what makes one think that

is actually the projection of

which is

and not just any other point on the line

?

Because

why is this not a projection of point

? Why do only

that satisfies this

have to be the projection of

?

I also found the answer.

Because: only for