Let \(V \subset \mathbb{R}^{n}\) be a subspace and \(Z \in \mathbb{R}^{n}\) a given vector. Find a unit vector \(X\) that is perpendicular to \(V\) with \(\langle X, Z\rangle\) as large as possible.

Question

a) Let \(V \subset \mathbb{R}^{n}\) be a subspace and \(Z \in \mathbb{R}^{n}\) a given vector. Find a unit vector \(X\) that is perpendicular to \(V\) with \(\langle X, Z\rangle\) as large as possible.
b) Compute max \(\int_{-1}^{1} x^{3} h(x) d x\) where \(h(x)\) is any continuous function on the interval \(-1 \leq x \leq 1\) subject to the restrictions
\[
\int_{-1}^{1} h(x) d x=\int_{-1}^{1} x h(x) d x=\int_{-1}^{1} x^{2} h(x) d x=0 ; \quad \int_{-1}^{1}|h(x)|^{2} d x=1 .
\]
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c) Compute \(\min _{a, b, c} \int_{-1}^{1}\left|x^{3}-a-b x-c x^{2}\right|^{2} d x\).