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For certain polynomials $\mathbf{p}(t), \mathbf{q}(t)$, and $\mathbf{r}(t)$, say we are given the following table of inner products:

\begin{align}
\begin{array}{|c||c|c|c|}
\hline\langle,\rangle & \mathbf{p} & \mathbf{q} & \mathbf{r} \\
\hline \hline \mathbf{p} & 4 & 0 & 8 \\
\mathbf{q} & 0 & 1 & 0 \\
\mathbf{r} & 8 & 0 & 50 \\
\hline
\end{array}
\end{align}

For example, $\langle\mathbf{q}, \mathbf{r}\rangle=\langle\mathbf{r}, \mathbf{q}\rangle=0$. Let $E$ be the span of $\mathbf{p}$ and $\mathbf{q}$.
a) Compute $\langle\mathbf{p}, \mathbf{q}+\mathbf{r}\rangle$.
b) Compute $\|\mathbf{q}+\mathbf{r}\|$.
c) Find the orthogonal projection $\operatorname{Proj}_{E} \mathbf{r}$. [Express your solution as linear combinations of $\mathbf{p}$ and $\mathbf{q}$.]
d) Find an orthonormal basis of the span of $\mathbf{p}, \mathbf{q}$, and $\mathbf{r}$. [Express your results as linear combinations of $\mathbf{p}, \mathbf{q}$, and $\mathbf{r}$.]
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