91Ó°ÊÓ

Determine whether the vectors are orthogonal, parallel, or neither. Explain. $$\mathbf{u}=(-\sin \theta, \cos \theta, 1), \quad \mathbf{v}=(\sin \theta,-\cos \theta, 0)$$

Find the orthogonal projection of \(f\) onto \(g .\) Use the inner product in \(C[a, b]\) $$\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x$$ $$C[-1,1], \quad f(x)=x, \quad g(x)=1$$

Find the Fourier approximation with the specified order of the function on the interval \([0,2 \pi]\). \(f(x)=\pi-x, \quad\) third order

Let \(\mathbf{v}=\left(v_{1}, v_{2}\right)\) be a vector in \(R^{2} .\) Show that \(\left(v_{2},-v_{1}\right)\) is orthogonal to \(v,\) and use this fact to find two unit vectors orthogonal to the given vector. $$\mathbf{v}=(8,15)$$

Find the orthogonal projection of \(f\) onto \(g .\) Use the inner product in \(C[a, b]\) $$\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x$$ $$C[-\pi, \pi], \quad f(x)=\sin x, \quad g(x)=\cos x$$