How do you do this vector projection?

Question

For $\textbf{v} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix}$ and $\textbf{w} = \begin{pmatrix} 2 \\ -1 \\ 0 \end{pmatrix}$, compute $\text{proj}_{\textbf{w}} \textbf{v}$.

I don't know how to project vectors of this form... I can only do it in 2D. Is there a formula for this?

Answer 1

For any vector u projected onto any vector v, take the dot product of u with v "hat", the unit vector of v (found by v/|v|). This dot product gives the scalar projection, to find the vector projection, simply multiply this scalar magnitude by the unit direction v "hat" again.

So for this problem, normalize w by its magnitude to obtain w "hat", then dot v with w "hat" to obtain the scalar projection (the magnitude of the vector projection), then just write this scalar times the w "hat" direction vector to obtain the complete vector projection expression.