Question

Given B={U, V} as an orthogonal basis of 2-space and X=aU+bV. Prove a=(X•U)/|U|^2
The capitals represent vectors

Answer 1

All right.
Because U and V are orthogonal, U.V=0
Thus, if I take the component of X in the direction of U, call it X_U, then X_U's dot product with V is also zero since X_U is in the direction of U.
Thus, we can compute the component of X in the direction of U as the dot product of X and the unit vector in the direction of U, which is (X.U)/|U|. However, since U in not necessarily a unit vector, we need to adjust this number in order to find a. Since U is necessarily |U| times the unit vector in the direction of U, we know that the component of X in the direction of U must be |U| times the amount U must be multiplied by to achieve the projection of X onto U. Thus, we divide the earlier quantity for the component of X into U by |U|, and achieve that a=(X.U)/|U|^2.

I'm sorry If that's not terribly clear, I can fix it up if you don't understand.

Answer 2

You can safely ignore the first part of that. It shouldn't matter that they are orthogonal for this.

Answer 3

That's brilliant! Thanks a lot!!

Answer 4

No problem