Chapter 10

Find the magnitude of the given vector. $$ \langle 5,0,-12\rangle $$

\(11-14\) . Find an equation of a sphere with the given radius \(r\) and center \(C .\) $$ r=3 ; \quad C(-1,4,-7) $$

Two vectors a and b are given. (a) Find a vector perpendicular to both a and b. (b) Find a unit vector perpendicular to both a and b. $$ \mathbf{a}=3 \mathbf{j}+5 \mathbf{k}, \quad \mathbf{b}=-\mathbf{i}+2 \mathbf{k} $$

Find parametric equations for the line that passes through the points \(P\) and \(Q .\) $$ P(3,7,-5), \quad Q(7,3,-5) $$

Find (a) \(\mathbf{u} \cdot \mathbf{v}\) and (b) the angle between \(\mathbf{u}\) and \(\mathbf{v}\) to the nearest degree. $$ \mathbf{u}=\mathbf{i}+3 \mathbf{j}, \quad \mathbf{v}=4 \mathbf{i}-\mathbf{j} $$

Find the magnitude of the given vector. $$ \langle 3,5,-4\rangle $$

\(11-14\) . Find an equation of a sphere with the given radius \(r\) and center \(C .\) $$ r=\sqrt{6} ; \quad Q(3,-1,0) $$

The lengths of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) and the angle \(\theta\) between them are given. Find the length of their cross product, \(|\mathbf{a} \times \mathbf{b}|\). $$ |\mathbf{a}|=6, \quad|\mathbf{b}|=\frac{1}{2}, \quad \theta=60^{\circ} $$

\(9-18\) . Express the vector with initial point \(P\) and terminal point \(Q\) in component form. $$ P(3,2), \quad Q(8,9) $$

Find parametric equations for the line that passes through the points \(P\) and \(Q .\) $$ P(12,16,18), \quad Q(12,-6,0) $$

Find (a) \(\mathbf{u} \cdot \mathbf{v}\) and (b) the angle between \(\mathbf{u}\) and \(\mathbf{v}\) to the nearest degree. $$ \mathbf{u}=3 \mathbf{i}+4 \mathbf{j}, \quad \mathbf{v}=-2 \mathbf{i}-\mathbf{j} $$

Find the magnitude of the given vector. $$ \langle 1,-6,2 \sqrt{2}\rangle $$

\(11-14\) . Find an equation of a sphere with the given radius \(r\) and center \(C .\) $$ r=\sqrt{11} ; \quad \quad \quad(-10,0,1) $$

The lengths of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) and the angle \(\theta\) between them are given. Find the length of their cross product, \(|\mathbf{a} \times \mathbf{b}|\). $$ |\mathbf{a}|=4, \quad|\mathbf{b}|=5, \quad \theta=30^{\circ} $$

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

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. $$ \mathbf{n}=\langle 1,1,-1\rangle, \quad P(0,2,-3) $$

Determine whether the given vectors are perpendicular. $$ \mathbf{u}=\langle 6,4\rangle, \quad \mathbf{v}=\langle- 2,3\rangle $$

Find the vectors \(\mathbf{u}+\mathbf{v}, \mathbf{u}-\mathbf{v},\) and \(3 \mathbf{u}-\frac{1}{2} \mathbf{v}\) $$ \mathbf{u}=\langle 2,-7,3\rangle, \mathbf{v}=\langle 0,4,-1\rangle $$

\(15-18=\) Show that the equation represents a sphere, and find its center and radius. $$ x^{2}+y^{2}+z^{2}-10 x+2 y+8 z=9 $$

The lengths of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) and the angle \(\theta\) between them are given. Find the length of their cross product, \(|\mathbf{a} \times \mathbf{b}|\). $$ |\mathbf{a}|=10, \quad|\mathbf{b}|=10, \quad \theta=90^{\circ} $$

\(9-18\) . Express the vector with initial point \(P\) and terminal point \(Q\) in component form. $$ P(5,3), \quad Q(1,0) $$

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. $$ \mathbf{n}=\langle 3,2,0\rangle, \quad P(1,2,7) $$

Find the vectors \(\mathbf{u}+\mathbf{v}, \mathbf{u}-\mathbf{v},\) and \(3 \mathbf{u}-\frac{1}{2} \mathbf{v}\) $$ \mathbf{u}=\langle 0,1,-3\rangle, \mathbf{v}=\langle 4,2,0\rangle $$

Determine whether the given vectors are perpendicular. $$ \mathbf{u}=\langle 0,-5\rangle, \quad \mathbf{v}=\langle 4,0\rangle $$

\(15-18=\) Show that the equation represents a sphere, and find its center and radius. $$ x^{2}+y^{2}+z^{2}+4 x-6 y+2 z=10 $$

The lengths of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) and the angle \(\theta\) between them are given. Find the length of their cross product, \(|\mathbf{a} \times \mathbf{b}|\). $$ |\mathbf{a}|=0.12, \quad|\mathbf{b}|=1.25, \quad \theta=75^{\circ} $$

\(9-18\) . Express the vector with initial point \(P\) and terminal point \(Q\) in component form. $$ P(-1,3), \quad Q(-6,-1) $$

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. $$ \mathbf{n}=\left\langle 3,0,-\frac{1}{2}\right\rangle, \quad P(2,4,8) $$

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. $$ \mathbf{u}=\langle- 2,6\rangle, \quad \mathbf{v}=\langle 4,2\rangle $$

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

Find a vector that is perpendicular to the plane passing through the three given points. $$ P(0,1,0), Q(1,2,-1), R(-2,1,0) $$

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

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. $$ \mathbf{n}=\left\langle-\frac{2}{3},-\frac{1}{3}, 1\right\rangle, \quad P(-6,0,-3) $$

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. $$ \mathbf{u}=2 \mathbf{i}, \quad \mathbf{v}=-7 \mathbf{j} $$

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

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

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

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. $$ \mathbf{n}=3 \mathbf{i}-\mathbf{j}+2 \mathbf{k}, \quad P(0,2,-3) $$

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

Determine whether the given vectors are perpendicular. $$ \mathbf{u}=2 \mathbf{i}-8 \mathbf{j}, \quad \mathbf{v}=-12 \mathbf{i}-3 \mathbf{j} $$

Describe the trace of the sphere $$ (x+1)^{2}+(y-2)^{2}+(z+10)^{2}=100 $$ in (a) the \(y\) z-plane and (b) the plane \(x=4\)

Find a vector that is perpendicular to the plane passing through the three given points. $$ P(1,1,-5), Q(2,2,0), R(0,0,0) $$

\(19-22\) . Sketch the given vector with initial point \((4,3),\) and find the terminal point. $$ \mathbf{u}=\langle 2,4\rangle $$

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. $$ \mathbf{n}=\mathbf{i}+4 \mathbf{j}, \quad P(1,0,-9) $$

Express the given vector in terms of the unit vectors i, j, and k. $$ \langle 0,-3,5\rangle $$

Determine whether the given vectors are perpendicular. $$ \mathbf{u}=4 \mathbf{i}, \quad \mathbf{v}=-\mathbf{i}+3 \mathbf{j} $$

Describe the trace of the sphere $$ x^{2}+(y-4)^{2}+(z-3)^{2}=144 $$ in (a) the \(x\) -plane and in (b) the plane \(z=-2\)

Find a vector that is perpendicular to the plane passing through the three given points. $$ P(3,0,0), Q(0,2,-5), R(-2,0,6) $$