Question

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.]