I was a bit worried if I followed all the conventions correctly or not.
Let the vector projection of a vector $\vec{a}$ on a non-zero vector $\vec{b}$ is $\vec{a_1}$.
My attempt:
$$\vec{a_1}=\frac{(\vec{a}\cdot\vec{b})\hat{b}}{|\vec{b}|}\tag{1}$$
$$=\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}\cdot\frac{\vec{b}}{|\vec{b}|}\tag{2}$$
$$=\frac{(\vec{a}\cdot\vec{b})\vec{b}}{|\vec{b}|^2}\tag{3}$$
Since $(\vec{a}\cdot \vec{b})$ is a scalar, placing $\hat{b}$ next to it to indicate multiplication shouldn't be a problem. Moreover, since $\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}$ is a scalar, putting "$\cdot$" between it and $\frac{\vec{b}}{|\vec{b}|}$ in $(2)$ shouldn't be a problem. Finally, as $(\vec{a}\cdot \vec{b})$ is a scalar, placing $\vec{b}$ next to it to indicate multiplication in shouldn't be a problem either. Also, in the wikipedia image below $\mathbf{\left(a\cdot b\right)\hat{b}}$ is written, so we can be certain that writing $(1)$ and $(3)$ is correct.
Question:
- Did I do everything right in $(2)$? I'm worried about the $\cdot$ between $\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}$ and $\frac{\vec{b}}{|\vec{b}|}$ in $(2)$.
Something similar from Wikipedia:
Expression $(2)$ is technically correct and ultimately unambiguous, but $$\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}\times\frac{\vec{b}}{|\vec{b}|}\tag{2r}$$ is better, and faster to parse, as it doesn't involve any double-checking that the author has indeed used “$\cdot$” to alternately mean dot product and regular multiplication.