91Ó°ÊÓ

\(23-24\) Determine whether the given vectors are orthogonal, parallel, or neither. $$ \begin{array}{l}{\text { (a) } \mathbf{a}=\langle- 5,3,7\rangle, \quad \mathbf{b}=\langle 6,-8,2\rangle} \\ {\text { (b) } \mathbf{a}=\langle 4,6\rangle, \quad \mathbf{b}=\langle- 3,2\rangle} \\ {\text { (c) } \mathbf{a}=-\mathbf{i}+ 2 \mathbf{j}+5 \mathbf{k}, \quad \mathbf{b}=3 \mathbf{i}+4 \mathbf{j}-\mathbf{k}} \\ {\text { (d) } \mathbf{a}=2 \mathbf{i}+6 \mathbf{j}-4 \mathbf{k}, \quad \mathbf{b}=-3 \mathbf{i}-9 \mathbf{j}+6 \mathbf{k}}\end{array} $$

Find an equation of the plane. The plane through the point \((6,3,2)\) and perpendicular to the vector \(\langle- 2,1,5\rangle\)

\(23-32\) Describe in words the region of \(\mathbb{R}^{3}\) represented by the equation or inequality. $$y=-4$$

Find a unit vector that has the same direction as the given vector. $$8 \mathbf{i}-\mathbf{j}+4 \mathbf{k}$$

Find an equation of the plane. The plane through the point \((4,0,-3)\) and with normal vector \(\mathbf{j}+2 \mathbf{k}\)

\(23-32\) Describe in words the region of \(\mathbb{R}^{3}\) represented by the equation or inequality. $$x=10$$

\(23-24\) Determine whether the given vectors are orthogonal, parallel, or neither. $$ \begin{array}{l}{\text { (a) } \mathbf{u}=\langle- 3,9,6\rangle, \quad \mathbf{v}=\langle 4,-12,-8\rangle} \\ {\text { (b) } \mathbf{u}=\mathbf{i}-\mathbf{j}+ 2 \mathbf{k}, \quad \mathbf{v}=2 \mathbf{i}-\mathbf{j}+\mathbf{k}} \\ {\text { (c) } \mathbf{u}=\langle a, b, c\rangle, \quad \mathbf{v}=\langle- b, a, 0\rangle}\end{array} $$

Find a vector that has the same direction as \(\langle- 2,4,2\rangle\) but has length \( 6 .\)

\(23-32\) Describe in words the region of \(\mathbb{R}^{3}\) represented by the equation or inequality. $$x>3$$

Find an equation of the plane. The plane through the point \((1,-1,1)\) and with normal vector \(\mathbf{i}+\mathbf{j}-\mathbf{k}\)