Chapter 9: Problem 4
Find the coordinates of the point. The point is located in the \(y z\) -plane, three units to the right of the \(x z\) -plane, and two units above the \(x y\) -plane.
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Determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal parallel, or neither. $$ \begin{array}{l} \mathbf{u}=\langle\cos \theta, \sin \theta,-1\rangle \\\\\mathbf{v}=\langle\sin \theta,-\cos \theta, 0\rangle \end{array} $$
Find the vector \(z,\) given that \(\mathbf{u}=\langle 1,2,3\rangle\) \(\mathbf{v}=\langle 2,2,-1\rangle,\) and \(\mathbf{w}=\langle 4,0,-4\rangle\) \(\mathbf{z}=2 \mathbf{u}+4 \mathbf{v}-\mathbf{w}\)
Find \(u \cdot v\). \(\|\mathbf{u}\|=40,\|\mathbf{v}\|=25,\) and the angle between \(\mathbf{u}\) and \(\mathbf{v}\) is \(5 \pi / 6\).
(a) find the projection of \(\mathbf{u}\) onto \(\mathbf{v}\), and (b) find the vector component of u orthogonal to v. $$ \mathbf{u}=\langle 2,-3\rangle, \quad \mathbf{v}=\langle 3,2\rangle $$
In Exercises 61 and \(62,\) sketch the solid that has the given description in cylindrical coordinates. $$ 0 \leq \theta \leq 2 \pi, 2 \leq r \leq 4, z^{2} \leq-r^{2}+6 r-8 $$
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