Orthogonal Complement question

Question

Let →a = ⟨−5,1⟩ and →b=⟨10,−1⟩

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

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I was able to find the projection which was ⟨-5.05, 0.505⟩ but I'm having trouble solving for the orthogonal complement. Thank you for your help :)

Answer 1

In order to find the orthogonal complement of a vector. You take the original vector and subtract it from the projection of the vector onto another vector. In other words, you subtract a parallel vector from the original vector to yield a perpendicular vector. Applying this logic, we will take our vector an and subtract it from the given projection. That is, <-5, 1> - <-5.05, 0.505> = <-1.05, 0.495>. This is our vector that is the orthogonal complement to our vector a.