91Ó°ÊÓ

For the three-dimensional vectors \(\mathbf{u}\) and \(\mathbf{v}\) in Problems 13-16, find the sum \(\mathbf{u}+\mathbf{v},\) the difference \(\mathbf{u}-\mathbf{v},\) and the magnitudes \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\| .\) $$ \mathbf{u}=\langle 0,0,0\rangle, \mathbf{v}=\langle-3,3,1\rangle $$

Name and sketch the graph of each of the following equations in three-space. $$x^{2}+y^{2}-4 z^{2}+4=0$$

Find a Cartesian equation of the plane through the three points \((2,3,-1),(-1,5,2),\) and (-4,-2,2)

. If \(\mathbf{u}+\mathbf{v}\) is orthogonal to \(\mathbf{u}-\mathbf{v},\) what can you say about the relative magnitudes of \(\mathbf{u}\) and \(\mathbf{v} ?\)

Sketch the graph of the given cylindrical or spherical equation. \(\rho=\sec \phi\)

Find \(\mathbf{r}^{\prime}(t)\) and \(\mathbf{r}^{\prime \prime}(t)\) for each of the following: (a) \(\mathbf{r}(t)=\left(e^{t}+e^{-l^{2}}\right) \mathbf{i}+2^{t} \mathbf{j}+t \mathbf{k}\) (b) \(\mathbf{r}(t)=\tan 2 t \mathbf{i}+\arctan t \mathbf{j}\)

Find the symmetric equations of the line through (-5,7,-2) and perpendicular to both \(\langle 2,1,-3\rangle\) and \(\langle 5,4,-1\rangle\)

In Problems 13-16, complete the squares to find the center and radius of the sphere whose equation is given (see Example 2). $$ x^{2}+y^{2}+z^{2}+2 x-6 y-10 z+34=0 $$

For the three-dimensional vectors \(\mathbf{u}\) and \(\mathbf{v}\) in Problems 13-16, find the sum \(\mathbf{u}+\mathbf{v},\) the difference \(\mathbf{u}-\mathbf{v},\) and the magnitudes \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\| .\) $$ \mathbf{u}=\langle 1,0,1\rangle, \mathbf{v}=\langle-5,0,0\rangle $$

Find parametric equations for the line through (-2,1,5) and (6,2,-3)