91Ó°ÊÓ

\(15-18=\) Show that the equation represents a sphere, and find its center and radius. $$ x^{2}+y^{2}+z^{2}=12 x+2 y $$

Find the vectors \(\mathbf{u}+\mathbf{v}, \mathbf{u}-\mathbf{v},\) and \(3 \mathbf{u}-\frac{1}{2} \mathbf{v}\) $$ \mathbf{u}=\mathbf{i}+\mathbf{j}, \mathbf{v}=-\mathbf{j}-2 \mathbf{k} $$

Determine whether the given vectors are perpendicular. $$ \mathbf{u}=\langle- 2,6\rangle, \quad \mathbf{v}=\langle 4,2\rangle $$

A plane has normal vector \(\mathbf{n}\) and passes through the point \(P\) (a) Find an equation for the plane. (b) Find the intercepts and sketch a graph of the plane. $$ \mathbf{n}=\left\langle-\frac{2}{3},-\frac{1}{3}, 1\right\rangle, \quad P(-6,0,-3) $$

\(9-18\) . Express the vector with initial point \(P\) and terminal point \(Q\) in component form. $$ P(-8,-6), \quad Q(-1,-1) $$

Find the vectors \(\mathbf{u}+\mathbf{v}, \mathbf{u}-\mathbf{v},\) and \(3 \mathbf{u}-\frac{1}{2} \mathbf{v}\) $$ \mathbf{u}=\langle a, 2 b, 3 c\rangle, \mathbf{v}=\langle- 4 a, b,-2 c\rangle $$

Determine whether the given vectors are perpendicular. $$ \mathbf{u}=2 \mathbf{i}, \quad \mathbf{v}=-7 \mathbf{j} $$

Find a vector that is perpendicular to the plane passing through the three given points. $$ P(3,4,5), Q(1,2,3), R(4,7,6) $$

\(15-18=\) Show that the equation represents a sphere, and find its center and radius. $$ x^{2}+y^{2}+z^{2}=14 y-6 z $$

Express the given vector in terms of the unit vectors i, j, and k. $$ \langle 12,0,2\rangle $$