91Ó°ÊÓ

Find an equation of the plane.The plane passes through \((1,2,3),(3,2,1)\), and \((-1,-2,2)\).

Find the following. (a) \(\|\mathbf{u}\|\) (b) \(\|\mathbf{v}\|\) (c) \(\|\mathbf{u}+\mathbf{v}\|\) (d) \(\quad \frac{\mathbf{u}}{\|\mathbf{u}\|}\) (e) \(\left\|\frac{\mathbf{v}}{\|\mathbf{v}\|}\right\|\) (f) \(\| \frac{\mathbf{u}+\mathbf{v}}{\|\mathbf{u}+\mathbf{v}\|}\) $$ \begin{aligned} &\mathbf{u}=\left\langle 1, \frac{1}{2}\right\rangle \\ &\mathbf{v}=\langle 2,3\rangle \end{aligned} $$

Complete the square to write the equation of the sphere in standard form. Find the center and radius. \(9 x^{2}+9 y^{2}+9 z^{2}-6 x+18 y+1=0\)

Find \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})\) $$ \begin{aligned} &\mathbf{u}=\langle 2,0,1\rangle \\ &\mathbf{v}=\langle 0,3,0\rangle \\ &\mathbf{w}=\langle 0,0,1\rangle \end{aligned} $$

Find an equation of the plane.The plane passes through the point \((1,2,3)\) and is parallel to the \(y z\) -plane.

Complete the square to write the equation of the sphere in standard form. Find the center and radius. \(4 x^{2}+4 y^{2}+4 z^{2}-4 x-32 y+8 z+33=0\)

Find the following. (a) \(\|\mathbf{u}\|\) (b) \(\|\mathbf{v}\|\) (c) \(\|\mathbf{u}+\mathbf{v}\|\) (d) \(\quad \frac{\mathbf{u}}{\|\mathbf{u}\|}\) (e) \(\left\|\frac{\mathbf{v}}{\|\mathbf{v}\|}\right\|\) (f) \(\| \frac{\mathbf{u}+\mathbf{v}}{\|\mathbf{u}+\mathbf{v}\|}\) $$ \begin{aligned} &\mathbf{u}=\langle 2,-4\rangle \\ &\mathbf{v}=\langle 5,5\rangle \end{aligned} $$

Find \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})\) $$ \begin{aligned} &\mathbf{u}=\langle 2,0,0\rangle \\ &\mathbf{v}=\langle 1,1,1\rangle \\ &\mathbf{w}=\langle 0,2,2\rangle \end{aligned} $$

Find an equation in spherical coordinates for the equation given in rectangular coordinates. \(x^{2}+y^{2}-3 z^{2}=0\)

Find the component of u that is orthogonal to \(v\), given \(w_{1}=\operatorname{proj}_{v} u\). $$ \mathbf{u}=\langle 9,7\rangle, \quad \mathbf{v}=\langle 1,3\rangle, \quad \text { proj } \mathbf{u}=\langle 3,9\rangle $$