Do we have $proj_u(a) + proj_u(b) = proj_u(a+b)$?

Question

Let $a, b, u$ be vectors in $\mathbb{R}^3$. For two vectors $r, u$ in $\mathbb{R}^3$, let $proj_u(r)$ be the projection of $r$ on the line of $u$ in $\mathbb{R}^3$. Do we have $proj_u(a) + proj_u(b) = proj_u(a+b)$? Thank you very much.

Answer 1

Usually we speak of $u$ as a vector in $\mathbb{R}^3$. It could also be pointing the the direction of a line.

$$proj_u(a) = \frac{a\cdot u}{\|u\|^2} \text{ and } proj_u(b) = \frac{b \cdot u}{\|u\|^2}$$

$$proj_u(a) + proj_u(b) = \frac{a\cdot u + b \cdot u}{\|u\|^2} = \frac{(a+b)\cdot u}{\|u\|^2} = proj_u(a+b)$$

Thus by linearity (i.e. the distributive property) of the dot product we have linearity of the projection.