Exercise 11.2.5

Suppose $B=\left\{\mathbf{u}_{1},\mathbf{u}_{2},\mathbf{u}_{3}\right\}$ is an orthogonal basis for an inner product space $V$, such that $\left\|\mathbf{u}_{1}\right\|=2$, $\left\|\mathbf{u}_{2}\right\|=\sqrt{3}$, and $\left\|\mathbf{u}_{3}\right\|=\sqrt{5}$. Moreover, suppose that $\mathbf{v}\in V$ is a vector such that $\langle\mathbf{v},\mathbf{u}_{1}\rangle=1$, $\langle\mathbf{v},\mathbf{u}_{2}\rangle=2$, and $\langle\mathbf{v},\mathbf{u}_{3}\rangle=-4$. Find the coordinates of $\mathbf{v}$ with respect to $B$.