Chapter 10.2, Problem 29P

a.

To determine

To show: That $u=a_1{v}_1+a_2{v}_2+a_3{v}_3$ for a set of 3 mutually orthogonal vectors in three dimensions $v_1,v_2,v_3$ and an arbitrary three-dimensional vector u with $a_i=\frac{u\cdot v_i}{v_i\cdot v_i},i=1,2,3$. Also, show that the coefficient $a_i$ can be interpreted as the projection of u in the direction of $v_i$ divided by the length of $v_i$.

b.

To determine

To show: That the equation ${\displaystyle {\int }_{-L}^{L}f\left(x\right)\cos \left(\frac{n\pi x}{L}\right)}dx=\left\{\begin{array}{l}\frac{a_0}{2}{\displaystyle {\int }_{-L}^L\cos \left(\frac{n\pi x}{L}\right)}dx+{\displaystyle \sum _{m=1}^{\infty }{a}_m{\displaystyle {\int }_{-L}^L\cos \left(\frac{m\pi x}{L}\right)\cos \left(\frac{n\pi x}{L}\right)}dx}+\\ {\displaystyle \sum _{m=1}^{\infty }{b}_m{\displaystyle {\int }_{-L}^L\sin \left(\frac{m\pi x}{L}\right)\cos \left(\frac{n\pi x}{L}\right)}dx}\end{array}\right\}$ can be written in the form $\left(f,{\varphi }_n\right)=\frac{a_0}{2}\left({\varphi }_0,{\varphi }_n\right)+{\displaystyle \sum _{m=1}^{\infty }{a}_m\left({\varphi }_m,{\varphi }_n\right)}+{\displaystyle \sum _{m=1}^{\infty }{b}_m\left({\psi }_m,{\varphi }_n\right)}$.

c.

To determine

To show: That the expression for $a_n$ is $a_n=\frac{\left(f,{\varphi }_n\right)}{\left({\varphi }_n,{\varphi }_n\right)}\text{;}n=0,1,2,\cdots$ and for $b_n$ is $b_n=\frac{\left(f,{\psi }_n\right)}{\left({\psi }_n,{\psi }_n\right)};n=1,2,\cdots$.