derivation of projection formula
Answers
Answer:
A unit vector is a vector of module one, which is given by the vector divided by its module. The vector projection of a vector on a vector other than zero b (also known as vector component or vector resolution of a in the direction of b) is the orthogonal projection of a on a straight line parallel to b.
Answer:
The derivation of the well-known projection formula projb⃗ (a⃗ )=a⃗ ⋅b⃗ b⃗ ⋅b⃗ b⃗ uses an argument based completely on geometry. We assume vectors are arrows in a normal cartesian space and use the law of cosines (among other things).
But now let's say that we have a vector space made up of all polynomials of order < N, and call our inner product
[a(x),b(x)]=∫1−1ab dx
where a(x) and b(x) are in polynomials in our vector space.
Okay, easy enough. But now things are getting complicated: the idea of a projection is a bit different. Now when we say projb(x)a(x) we mean to say "write a(x) as the linear combination of the two vectors b(x) and "something else", and give back only the part with b(x)". Obviously this "something else" is a polynomial that is orthogonal to b(x)....
....except that this isn't obvious at all. How can we say that just because the inner product of b(x) and "something else" is 0 that they have no "common directionality"? Even worse, what does "directionality" even mean?
So this is my question: "How can we derive the projection formula without making references to geometry?"
Thank you very much