Vector Product: Why is $\sin\theta$ equal to this?

Question

In my textbook it defines the vector cross product as:

$a \times b = |a||b|\sin\theta \hat n$

But then $\sin\theta$ is:

$\sin\theta= \frac{|a\times b|}{|a||b|}$

Where does this come from? and why can $\hat n$ just be 'ignored'?

Answer 1

In the first formula $\hat n$ is supposed to be a common normal vector to $a$ and $b$. One of the things this means is that $\hat n$ is by definition expected to have unit length. So if you take the length of both sides of the first equation you get $$ |a\times b| = |a| |b| |\sin(\theta)| |\hat n| $$ but $|\hat n|=1$ by definition, so dividing by $|a||b|$ gives you $$ |\sin\theta| = \frac{|a\times b|}{|a||b|}$$ If $\theta$ is measured to be between $0$ and $\pi$, its sine cannot be negative, so the absolute signs on the left disappear.

Answer 2

$\hat n$ accounts for direction of vector $\vec a\times \vec b$ when you take magnitude of this $\vec a\times \vec b$ vector and use $|\hat n|=1$ then there will be no longer $\hat n$ present in the picture