Projecting vector onto perp of span

Question

Find the projection of the vector $v = \begin{bmatrix}1 &2 & 3 & 4\end{bmatrix}^T$ onto $\operatorname{Span} \left(\begin{bmatrix}1 & 1 & 1 &1\end{bmatrix}^T\right)^\perp.$

I understand how to project v onto the span, but I am confused how to approach the problem given perp.

Is this the same thing as taking the orthogonal compliment first and then projecting $v$ onto that?

Answer 1

Guide:

Suppose we want to compute projection of $v$ onto $\operatorname{span}(w)^\perp$.

We first compute projection $v$ onto $\operatorname{span}(w)$, call it $u$.

Our desired answer would be $v-u$.