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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.]
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