# Prove Cosine law using vectors!

• Jun 10th 2012, 05:16 PM
Clairvoyantski
Prove Cosine law using vectors!
If C (dot) C= IC^2I how can I prove cosine law with vectors?
• Jun 10th 2012, 05:46 PM
Reckoner
Re: Prove Cosine law using vectors!
Quote:

Originally Posted by Clairvoyantski
If C (dot) C= IC^2I how can I prove cosine law with vectors?

Use this property:

$\displaystyle \mathbf{u}\cdot\mathbf{v}=\left\|\mathbf{u} \right\|\left\|\mathbf{v}\right\|\cos\theta,$

where $\displaystyle \theta$ is the angle between $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$.

Form a triangle with $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$ as two of its sides. The remaining side is represented by $\displaystyle \mathbf{u}-\mathbf{v}$. Squaring the length of this side gives

$\displaystyle \left\|\mathbf{u}-\mathbf{v}\right\|^2$

$\displaystyle =\left(\mathbf{u}-\mathbf{v}\right)\cdot\left(\mathbf{u}-\mathbf{v}\right)$

$\displaystyle =\mathbf{u}\cdot\left(\mathbf{u}-\mathbf{v}\right)-\mathbf{v}\cdot\left(\mathbf{u}-\mathbf{v}\right)$

$\displaystyle =\mathbf{u}\cdot\mathbf{u}-\mathbf{u}\cdot\mathbf{v}-\mathbf{v}\cdot\mathbf{u}+\mathbf{v}\cdot\mathbf{v }\right)$

Now just finish it up.