91Ó°ÊÓ

Find the angle \(\theta\) between the vectors. $$ \begin{aligned} &\mathbf{u}=\cos \left(\frac{\pi}{6}\right) \mathbf{i}+\sin \left(\frac{\pi}{6}\right) \mathbf{j} \\ &\mathbf{v}=\cos \left(\frac{3 \pi}{4}\right) \mathbf{i}+\sin \left(\frac{3 \pi}{4}\right) \mathbf{j} \end{aligned} $$

The line passes through the point \((2,3,4)\) and is perpendicular to the plane given by \(3 x+2 y-z=6\)

Find an equation in cylindrical coordinates for the equation given in rectangular coordinates. \(x^{2}+y^{2}+z^{2}=10\)

The initial and terminal points of a vector \(\mathbf{v}\) are given. (a) Sketch the given directed line segment, (b) write the vector in component form, and (c) sketch the vector with its initial point at the origin. $$ \left(\frac{3}{2}, \frac{4}{3}\right) \quad\left(\frac{1}{2}, 3\right) $$

Find the angle \(\theta\) between the vectors. $$ \begin{aligned} &\mathbf{u}=\langle 1,1,1\rangle \\ &\mathbf{v}=\langle 2,1,-1\rangle \end{aligned} $$

Find \(\mathrm{u} \times \mathrm{v}\) and show that it is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\). $$ \begin{aligned} &\mathbf{u}=\mathbf{i}+\mathbf{j}+\mathbf{k} \\ &\mathbf{v}=2 \mathbf{i}+\mathbf{j}-\mathbf{k} \end{aligned} $$

Describe and sketch the surface. $$ z-\sin y=0 $$

Describe and sketch the surface. $$ z-e^{y}=0 $$

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

Determine the location of a point \((x, y, z)\) that satisfies the condition(s). \(z=-3\)