In this problem we indicate certain similarities between three dimensional
geometric vectors and Fourier series.
(a) Let \(\mathbf{v}_{1}, \mathbf{v}_{2},\) and \(\mathbf{v}_{3}\) be a set of
mutually orthogonal vectors in three dimensions and let \(\mathbf{u}\) be any
three-dimensional vector. Show that $$\mathbf{u}=a_{1} \mathbf{v}_{1}+a_{2}
\mathbf{v}_{2}+a_{3} \mathbf{v}_{3}$$ where $$a_{i}=\frac{\mathbf{u} \cdot
\mathbf{v}_{i}}{\mathbf{v}_{i} \cdot \mathbf{v}_{i}}, \quad i=1,2,3$$
Show that \(a_{i}\) can be interpreted as the projection of \(\mathbf{u}\) in the
direction of \(\mathbf{v}_{i}\) divided by the
length of \(\mathbf{v}_{i}\).
(b) Define the inner product \((u, v)\) by $$(u, v)=\int_{-L}^{L} u(x) v(x) d
x$$ Also let $$\begin{array}{ll}{\phi_{x}(x)=\cos (n \pi x / L),} & {n=0,1,2,
\ldots} \\ {\psi_{n}(x)=\sin (n \pi x / L),} & {n=1,2, \ldots}\end{array}$$
Show that Eq. ( 10 ) can be written in the form $$\left(f,
\phi_{n}\right)=\frac{a_{0}}{2}\left(\phi_{0},
\phi_{n}\right)+\sum_{m=1}^{\infty} a_{m}\left(\phi_{m},
\phi_{n}\right)+\sum_{m=1}^{\infty} b_{m}\left(\psi_{m}, \phi_{m}\right)$$
(c) Use Eq. (v) and the corresponding equation for \(\left(f, \psi_{n}\right)\)
together with the orthogonality relations to show that $$a_{n}=\frac{\left(f,
\phi_{n}\right)}{\left(\phi_{n}, \phi_{n}\right)}, \quad n=0,1,2, \ldots ;
\quad b_{n}=\frac{\left(f, \psi_{n}\right)}{\left(\psi_{n}, \psi_{n}\right)},
\quad n=1,2, \ldots$$
Note the resemblance between Eqs. (vi) and Eq. (ii). The functions \(\phi_{x}\)
and \(\psi_{x}\) play a role for functions similar to that of the orthogonal
vectors \(v_{1}, v_{2},\) and \(v_{3}\) in three-dimensional
