91影视

Find the distance between the point and the plane. $$ (1,-2,3) ; 2 x-2 y+z=4 $$

Find the distance between the point and the plane. $$ (0,1,5) ; 3 x+6 y-2 z-5=0 $$

These exercises refer to the hyperbolic paraboloid \(z=y^{2}-x^{2}\) (a) Find an equation of the hyperbolic trace in the plane \(z=-4\). (b) Find the vertices of the hyperbola in part (a). (c) Find the foci of the hyperbola in part (a). (d) Describe the orientation of the focal axis of the hyperbola in part (a) relative to the coordinate axes.

Sketch the surface in 3 -space. $$ y z=1 $$

Describe the line segment represented by the vector equation. $$ \langle x, y, z\rangle=\langle- 2,1,4\rangle+ t\langle 3,0,-1\rangle \quad(0 \leq t \leq 3) $$

An equation of a surface is given in rectangular coordinates. Find an equation of the surface in (a) cylindrical coordinates and (b) spherical coordinates. $$ x^{2}+y^{2}-z^{2}=1 $$

Let \(\mathbf{r}_{1}=\left\langle x_{1}, y_{1}\right\rangle, \mathbf{r}_{2}=\left\langle x_{2}, y_{2}\right\rangle,\) and \(\mathbf{r}=\langle x, y\rangle .\) Assuming that \(k>\left\|\mathbf{r}_{2}-\mathbf{r}_{1}\right\|,\) describe the set of all points \((x, y)\) for which \(\left\|\mathbf{r}-\mathbf{r}_{1}\right\|+\left\|\mathbf{r}-\mathbf{r}_{2}\right\|=k\)

Expressions of the form $$ \mathbf{u} \times(\mathbf{v} \times \mathbf{w}) \quad \text { and } \quad(\mathbf{u} \times \mathbf{v}) \times \mathbf{w} $$ are called vector triple products. It can be proved with some effort that $$ \begin{array}{l}{\mathbf{u} \times(\mathbf{v} \times \mathbf{w})=(\mathbf{u} \cdot \mathbf{w}) \mathbf{v}-(\mathbf{u} \cdot \mathbf{v}) \mathbf{w}} \\\ {(\mathbf{u} \times \mathbf{v}) \times \mathbf{w}=(\mathbf{w} \cdot \mathbf{u}) \mathbf{v}-(\mathbf{w} \cdot \mathbf{v}) \mathbf{u}}\end{array} $$ These expressions can be summarized with the follow- ing mnemonic rule: vector triple product \(=\) (outer \(\cdot\) remote)adjacent \(-\text { (outer } \cdot \text { adjacent })\) remote See if you can figure out what the expressions 鈥渙uter,鈥� 鈥渞emote,鈥� and 鈥渁djacent鈥� mean in this rule, and then use the rule to find the two vector triple products of the vectors $$ \mathbf{u}=\mathbf{i}+3 \mathbf{j}-\mathbf{k}, \quad \mathbf{v}=\mathbf{i}+\mathbf{j}+2 \mathbf{k}, \quad \mathbf{w}=3 \mathbf{i}-\mathbf{j}+2 \mathbf{k} $$

An equation of a surface is given in rectangular coordinates. Find an equation of the surface in (a) cylindrical coordinates and (b) spherical coordinates. $$ x^{2}=16-z^{2} $$

These exercises refer to the hyperbolic paraboloid \(z=y^{2}-x^{2}\) (a) Find an equation of the parabolic trace in the plane \(x=2 .\) (b) Find the vertices of the parabola in part (a). (c) Find the focus of the parabola in part (a). (d) Describe the orientation of the focal axis of the parabola in part (a) relative to the coordinate axes.