Vector Projection, and Orthogonal Complement

Question

I have projections down good, but I keep getting the orthogonal complement wrong. 

 

QUESTION

 

a = <0,2,-5>     b = <6,8,-3>

 

compute the projection of a onto b and the vector component of a orthogonal to b. 

Answer 1

In general, a vector a can be decomposed into two parts: a component parallel to some other vector b and a component orthogonal to b. That is, if a_|| is the component of a parallel to b and a_⊥ is the component orthogonal to b, then a = a_|| + a_⊥. So to find the vector component of a orthogonal to b, you just need to subtract the component parallel to b from a:

 

a_⊥ = a - a_||