How do I deal with division by zero when simplifying equations?

Question

Can I simplify following equation with vectors $$ \frac{\vec v \cdot \vec w}{\vec w \cdot \vec w}\vec w = \vec v $$ to following and how? $$ \vec v = \vec v $$

I'm stuck trying to figure out how to handle division by zero. $\vec w \cdot \vec w = 0$, right? I got as far as the following, but can't figure out how to proceed.

$$ \frac{\vec v \cdot \vec w}{0}\vec w = \vec v $$

Answer 1

First, $\vec w \cdot \vec w$ is the squared length of $\vec w$. It is only $0$ if $\vec w$ is the zero vector.

Your simplification is not correct. The left side is in the direction of $\vec w$ while the right side is in the direction of $\vec v$. Let $\vec v=(1,0), \vec w=(1,2)$. Then $$\vec w \cdot \vec w=(1,2)\cdot (1,2)=1\cdot 1+2\cdot 2=5\\ \vec v \cdot \vec w=(1,0)\cdot (1,2)=1\\ \frac{\vec v \cdot \vec w}{\vec w \cdot \vec w}\vec w = \frac 15\vec w=(\frac 15,\frac 25)\\ \vec v=(1,0)$$

Answer 2

Careful, $\vec{w} \cdot \vec{w}=|w|^2$, not $0$. And so $\vec{w} \cdot \vec{w}=0$ if and only if $\vec{w} = \vec{0}$.