Let \(U, V, W\) be orthogonal vectors and let \(Z=a U+b V+c W\), where \(a, b, c\) are scalars.

a) (Pythagoras) Show that \(\|Z\|^{2}=a^{2}\|U\|^{2}+b^{2}\|V\|^{2}+c^{2}\|W\|^{2}\).

b) Find a formula for the coefficient \(a\) in terms of \(U\) and \(Z\) only. Then find similar formulas for \(b\) and \(c\). [Suggestion: take the inner product of \(Z=a U+b V+c W\) with \(U\) ].

REMARK The resulting simple formulas are one reason that orthogonal vectors are easier to use than more general vectors. This is vital for Fourier series.

c) Solve the following equations:

\[

\begin{aligned}

&x+y+z+w=2 \\

&x+y-z-w=3 \\

&x-y+z-w=0 \\

&x-y-z+w=-5

\end{aligned}

\]

[Suggestion: Observe that the columns vectors in the coefficient matrix are orthogonal.]