Prove Cosine law using vectors!

**Clairvoyantski** · June 10th 2012, 05:16 PM

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

**Reckoner** · June 10th 2012, 05:46 PM

Originally Posted by Clairvoyantski

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

Use this property:

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

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

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

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

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

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

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

Now just finish it up.

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