Difference of inner product space of two vectors

Question

If in an inner product space $\alpha,\beta$ are two vectors such that $\|\alpha\|= 2,\|\beta\|=3$, and $\|\alpha+\beta\|=5$. Then $\|\alpha-\beta\|$ is equal to ?

The options are 1)0 2)1 3)√10 4)√12

Answer 1

There is a general connection between the inner product and the norm called the polarization identity.

First, every inner product space has an induced norm which satisfies the parallelogram law: $$2\|\mathbf{u}\|^2 + 2\|\mathbf{v}\|^2 = \|\mathbf{u}+\mathbf{v}\|^2 + \|\mathbf{u}-\mathbf{v}\|^2.$$ From this, you can find $\|\mathbf{u}-\mathbf{v}\|$, given $\|\mathbf{u}\|$, $\|\mathbf{v}\|$, and $\|\mathbf{u}+\mathbf{v}\|$.

Conversely, every normed vector space with a norm that satisfies the parallelogram law has an induced inner product given by the following polarization identity (over $\mathbb{R}$, with some slight modifications for $\mathbb{C}$) $$\langle \mathbf{u},\mathbf{v}\rangle = \frac{1}{4}\left(\|\mathbf{u}+\mathbf{v}\|^2 - \|\mathbf{u}-\mathbf{v}\|^2\right).$$ This will allow you find $\langle\mathbf{u},\mathbf{v}\rangle$ for any vectors $\mathbf{u},\mathbf{v}$ given the norms $\|\mathbf{u}+\mathbf{v}\|$ and $\|\mathbf{u}-\mathbf{v}\|$.