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.
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 ?
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.
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 ?