91Ó°ÊÓ

Find vector \(\mathbf{c} \times(\mathbf{a}+3 \mathbf{b})\), where \(\mathbf{a}=\left|\begin{array}{lll}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ 5 & 0 & 9 \\ 0 & 1 & 0\end{array}\right|, \mathbf{b}=\left|\begin{array}{rrr}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ 0 & -1 & 1 \\ 7 & 1 & -1\end{array}\right|\), and \(\mathbf{c}=\mathbf{i}-\mathbf{k}\).

Use the cross product \(\mathbf{u} \times \mathbf{v}\) to find the acute angle between vectors \(\mathbf{u}\) and \(\mathbf{v}\), where \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}\) and \(\mathbf{v}=\mathbf{i}+\mathbf{k}\). Express the answer in degrees rounded to the nearest integer.

For the following exercises determine whether the given vectors are orthogonal. $$ \mathbf{a}=3 \mathbf{i}-\mathbf{j}-2 \mathbf{k}, \mathbf{b}=-2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} $$

Let \(\mathbf{v}=\langle 1,2\rangle\) a. Find a unit vector with the same direction as \(\mathbf{v}\). b. Find a vector \(\mathbf{w}\) with the same direction as \(\mathbf{v}\) such that \(\|\mathbf{w}\|=7\).

Rewrite the given equation of the quadric surface in standard form. Identify the surface. $$ 5 y=x^{2}-z^{2} $$

Find the measure of the angle between planes \(x+y-z=3\) and \(3 x-y+3 z=5\) Give the answer in radians and round to two decimal places.

For the following exercises, the rectangular coordinates \((x, y, z)\) of a point are given. Find the cylindrical coordinates \((r, \theta, z)\) of the point.\((-2 \sqrt{2}, 2 \sqrt{2}, 4)\)

Find terminal point \(Q\) of vector \(\overrightarrow{P Q}=\langle 7,-1,3\rangle\) with the initial point at \(P(-2,3,5)\).

For the following exercises determine whether the given vectors are orthogonal. $$ \mathbf{a}=\mathbf{i}-\mathbf{j}, \mathbf{b}=7 \mathbf{i}+2 \mathbf{j}-\mathbf{k} $$

For the following exercises, the equation of a surface in cylindrical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface.[7] \(r=4\)