# How do I find the sine of the angle between two vectors?

Jul 28, 2018

The sine of the angle between $\vec{u}$ and $\vec{v}$ is:

$\frac{\vec{u} \times \vec{v}}{\left\mid u \right\mid \left\mid v \right\mid}$

#### Explanation:

I will assume you mean real valued two dimensional vectors..

Given vectors, $\vec{u}$ and $\vec{v}$, note that they can be represented in polar form as:

$\vec{u} = \left\mid u \right\mid \left(\left(\cos \alpha\right) \hat{i} + \left(\sin \alpha\right) \hat{j}\right)$

$\vec{v} = \left\mid v \right\mid \left(\left(\cos \beta\right) \hat{i} + \left(\sin \beta\right) \hat{j}\right)$

where $\alpha$ and $\beta$ are the angles that $\vec{u}$ and $\vec{v}$ make with the $x$ axis.

Then:

$\vec{u} \times \vec{v} = \left\mid u \right\mid \left(\cos \alpha\right) \left\mid v \right\mid \left(\sin \beta\right) - \left\mid u \right\mid \left(\sin \alpha\right) \left\mid v \right\mid \left(\cos \beta\right)$

$\textcolor{w h i t e}{\vec{u} \times \vec{v}} = \left\mid u \right\mid \left\mid v \right\mid \left(\cos \alpha \sin \beta - \sin \alpha \cos \beta\right)$

$\textcolor{w h i t e}{\vec{u} \times \vec{v}} = \left\mid u \right\mid \left\mid v \right\mid \sin \left(\beta - \alpha\right)$

So:

$\sin \left(\beta - \alpha\right) = \frac{\vec{u} \times \vec{v}}{\left\mid u \right\mid \left\mid v \right\mid}$

which is the sine of the angle between the two vectors.

Three dimensions

For $3$ dimensional vectors $\vec{u}$ and $\vec{v}$, the cross product is a vector quantity rather than a scalar one, but the absolute value of the sine of the angle between $\vec{u}$ and $\vec{v}$ is expressible in terms of the length of that vector quantity as:

$\frac{\left\mid \vec{u} \times \vec{v} \right\mid}{\left\mid u \right\mid \left\mid v \right\mid}$