Exercise 11.1.4

For $\mathbf{u},\mathbf{v}$ vectors in $\mathbb{R}^{3}$, define the product $\mathbf{u}\ast\mathbf{v}=u_{1}v_{1}+2u_{2}v_{2}+3u_{3}v_{3}$. Show that

Hint: first show that the operation $\langle\mathbf{u},\mathbf{v}\rangle=\mathbf{u}\ast\mathbf{v}$ is an inner product on $\mathbb{R}^{3}$, then use Proposition 11.13.