Question

Let $u$ and $v$ be two nonzero vectors in ${\mathbb{R}}^2$. Prove that $u\cdot v=\Vert u\Vert \Vert v\Vert cos\theta$ where $\theta$ is the angle between $u$ and $v$.

Short answer

It is proven that $u\cdot v=\Vert u\Vert \Vert v\Vert cos\theta$.

Step-by-step solution

Step 1. Prove

To prove that $u\cdot v=\Vert u\Vert \Vert v\Vert cos\theta$ where $\theta$ is the angle between $u$ and $v$ we will use the law of cosines and the properties of dot-product.

So, the law of cosines for the vectors $u$ and $v$ is

${\Vert u-v\Vert }^2={\Vert u\Vert }^2+{\Vert v\Vert }^2-2\Vert u\Vert \Vert v\Vert cos\theta$ .....(a)

Step 2. Prove

Now, by using the dot-product we get,

${\Vert u-v\Vert }^2=(u-v)\cdot (u-v)$

${\Vert u-v\Vert }^2=u\cdot u-u\cdot v+v\cdot u+v\cdot v$

${\Vert u-v\Vert }^2=u\cdot u-2u\cdot v+v\cdot v$

${\Vert u-v\Vert }^2={\Vert u\Vert }^2-2u\cdot v+{\Vert v\Vert }^2$ .......(b)

By using equations (a) and (b)

${\Vert u\Vert }^2-2u\cdot v+{\Vert v\Vert }^2={\Vert u\Vert }^2+{\Vert v\Vert }^2-2\Vert u\Vert \Vert v\Vert cos\theta$

$-2u\cdot v=-2\Vert u\Vert \Vert v\Vert cos\theta$

$u\cdot v=\Vert u\Vert \Vert v\Vert cos\theta$

Hence proved.