91Ó°ÊÓ

Find an equation of the plane passing through points \((1,9,2),(1,3,6)\), and \((1,-7,8)\).

For the following exercises, the two-dimensional vectors a and b are given. Find the measure of the angle ? between a and b. Express the answer in radians rounded to two decimal places, if it is not possible to express it exactly. Is ? an acute angle? $$ \mathbf{a}=\langle 2,1\rangle, \mathbf{b}=\langle-1,3\rangle $$

Describe the relationship between the lines with the following parametric equations: $$ \begin{aligned} &x=1-4 t, y=3+t, z=8-6 t \\ &x=2+3 s, y=2 s, z=-1-3 s \end{aligned} $$

Identifying Surfaces in the Spherical Coordinate System Describe the surfaces with the given spherical equations. a. \(\theta=\frac{\pi}{3}\) b. \(\varphi=\frac{5 \pi}{6}\) c. \(\rho=6\) d. \(\rho=\sin \theta \sin \varphi\)

Vectors \(\mathbf{u}\) and \(\mathbf{v}\) are given. Find unit vector \(\mathbf{w}\) in the direction of the cross product vector \(\mathbf{u} \times \mathbf{v}\). Express your answer using standard unit vectors. $$ \mathbf{u}=\overrightarrow{O P}, \mathbf{v}=\overrightarrow{P Q}, \text { where } P(-1,1,0) \text { and } Q(0,2,1) $$

For the following exercises, find the equation of the sphere in standard form that satisfies the given conditions. Center \(C(-1,7,4)\) and radius 4

The vector with initial point \((0,0)\) and terminal point \((x, y)\) can be written in component form as $$ \mathbf{v}=\langle x, y\rangle $$ The scalars \(x\) and \(y\) are called the components of \(\mathbf{v}\).

For the following exercises, the two-dimensional vectors a and b are given. Find the measure of the angle ? between a and b. Express the answer in radians rounded to two decimal places, if it is not possible to express it exactly. Is ? an acute angle? $$ \mathbf{u}=3 \mathbf{i}, \mathbf{v}=4 \mathbf{i}+4 \mathbf{j} $$

Match the given quadric surface with its corresponding equation in standard form. Hyperboloid of two sheets

Determine the real number \(\alpha\) such that \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{i}\) are orthogonal, where \(\mathbf{u}=3 \mathbf{i}+\mathbf{j}-5 \mathbf{k}\) and \(\mathbf{v}=4 \mathbf{i}-2 \mathbf{j}+\alpha \mathbf{k}\).