Chapter 10

\(19-22\) . Sketch the given vector with initial point \((4,3),\) and find the terminal point. $$ \mathbf{u}=\langle- 1,2\rangle $$

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

Find an equation of the plane that passes through the points \(P, Q,\) and \(R .\) $$ P(6,-2,1), \quad Q(5,-3,-1), \quad R(7,0,0) $$

Find the indicated quantity, assuming \(\mathbf{u}=2 \mathbf{i}+\mathbf{j}, \mathbf{v}=\mathbf{i}-3 \mathbf{j},\) and \(\mathbf{w}=3 \mathbf{i}+4 \mathbf{j}\) $$ \mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w} $$

Spherical Water Tank A water tank is in the shape of a sphere of radius 5 feet. The tank is supported on a metal circle 4 feet below the center of the sphere, as shown in the figure. Find the radius of the metal circle.

Find the area of the parallelogram determined by the given vectors. $$ \mathbf{u}=\langle 3,2,1\rangle, \quad \mathbf{v}=\langle 1,2,3\rangle $$

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

Express the given vector in terms of the unit vectors i, j, and k. $$ \left\langle- a, \frac{1}{3} a, 4\right\rangle $$

Find an equation of the plane that passes through the points \(P, Q,\) and \(R .\) $$ P(3,4,5), \quad Q(1,2,3), \quad R(4,7,6) $$

Find the indicated quantity, assuming \(\mathbf{u}=2 \mathbf{i}+\mathbf{j}, \mathbf{v}=\mathbf{i}-3 \mathbf{j},\) and \(\mathbf{w}=3 \mathbf{i}+4 \mathbf{j}\) $$ \mathbf{u} \cdot(\mathbf{v}+\mathbf{w}) $$

A Spherical Buoy A spherical buoy of radius 2 feet floats in a calm lake. Six inches of the buoy are submerged. Place a coordinate system with the origin at the center of the sphere. (a) Find an equation of the sphere. (b) Find an equation of the circle formed at the waterline of the buoy.

Find the area of the parallelogram determined by the given vectors. $$ \mathbf{u}=\langle 0,-3,2\rangle, \quad \mathbf{v}=\langle 5,-6,0\rangle $$

\(19-22\) . Sketch the given vector with initial point \((4,3),\) and find the terminal point. $$ \mathbf{u}=\langle- 8,-1\rangle $$

Find an equation of the plane that passes through the points \(P, Q,\) and \(R .\) $$ P\left(3, \frac{1}{3},-5\right), \quad Q\left(4, \frac{2}{3},-3\right), \quad R(2,0,1) $$

Find the indicated quantity, assuming \(\mathbf{u}=2 \mathbf{i}+\mathbf{j}, \mathbf{v}=\mathbf{i}-3 \mathbf{j},\) and \(\mathbf{w}=3 \mathbf{i}+4 \mathbf{j}\) $$ (\mathbf{u}+\mathbf{v}) \cdot(\mathbf{u}-\mathbf{v}) $$

Visualizing a Set in Space Try to visualize the set of all points \((x, y, z)\) in a coordinate space that are equidistant from the points \(P(0,0,0)\) and \(Q(0,3,0) .\) Use the Distance Formula to find an equation for this surface, and observe that it is a plane.

Find the area of the parallelogram determined by the given vectors. $$ \mathbf{u}=2 \mathbf{i}-\mathbf{j}+4 \mathbf{k}, \quad \mathbf{v}=\frac{1}{2} \mathbf{i}+2 \mathbf{j}-\frac{3}{2} \mathbf{k} $$

\(23-26\) Sketch representations of the given vector with initial points at \((0,0),(2,3),\) and \((-3,5)\). $$ \mathbf{u}=\langle 3,5\rangle $$

Two vectors \(\mathbf{u}\) and \(\mathbf{v}\) are given. Express the vector \(-2 \mathbf{u}+3 \mathbf{v}(\mathbf{a})\) in component form \(\left\langle a_{1}, a_{2}, a_{3}\right\rangle\) and \((\mathbf{b})\) in terms of $$ \mathbf{u}=\langle 3,1,0\rangle, \quad \mathbf{v}=\langle 3,0,-5\rangle $$

Find an equation of the plane that passes through the points \(P, Q,\) and \(R .\) $$ P\left(\frac{3}{2}, 4,-2\right), \quad Q\left(-\frac{1}{2}, 2,0\right), \quad R\left(-\frac{1}{2}, 0,2\right) $$

Find the indicated quantity, assuming \(\mathbf{u}=2 \mathbf{i}+\mathbf{j}, \mathbf{v}=\mathbf{i}-3 \mathbf{j},\) and \(\mathbf{w}=3 \mathbf{i}+4 \mathbf{j}\) $$ (\mathbf{u} \cdot \mathbf{v})(\mathbf{u} \cdot \mathbf{w}) $$

Visualizing a Set in Space Try to visualize the set of all points \((X, y, z)\) in a coordinate space that are twice as far from the points \(Q(0,3,0)\) as from the point \(P(0,0,0) .\) Use the Distance Formula to show that the set is a sphere, and find its center and radius.

Find the area of the parallelogram determined by the given vectors. $$ \mathbf{u}=\mathbf{i}-\mathbf{j}+\mathbf{k}, \quad \mathbf{v}=\mathbf{i}+\mathbf{j}-\mathbf{k} $$

\(23-26\) Sketch representations of the given vector with initial points at \((0,0),(2,3),\) and \((-3,5)\). $$ \mathbf{u}=\langle 4,-6\rangle $$

Two vectors \(\mathbf{u}\) and \(\mathbf{v}\) are given. Find their dot product \(\mathbf{u} \cdot \mathbf{v} .\) $$ \mathbf{u}=\langle 2,5,0\rangle, \quad \mathbf{v}=\left\langle\frac{1}{2},-1,10\right\rangle $$

Find an equation of the plane that passes through the points \(P, Q,\) and \(R .\) $$ P(6,1,1), \quad Q(3,2,0), \quad R(0,0,0) $$

Find the component of \(\mathbf{u}\) along \(\mathbf{v}\) $$ \mathbf{u}=\langle 4,6\rangle, \quad \mathbf{v}=\langle 3,-4\rangle $$

Find the area of \(\triangle P Q R\) $$ P(1,0,1), Q(0,1,0), R(2,3,4) $$

\(23-26\) Sketch representations of the given vector with initial points at \((0,0),(2,3),\) and \((-3,5)\). $$ \mathbf{u}=\langle- 7,2\rangle $$

Two vectors \(\mathbf{u}\) and \(\mathbf{v}\) are given. Find their dot product \(\mathbf{u} \cdot \mathbf{v} .\) $$ \mathbf{u}=\langle- 3,0,4\rangle, \quad \mathbf{v}=\left\langle 2,4, \frac{1}{2}\right\rangle $$

Find an equation of the plane that passes through the points \(P, Q,\) and \(R .\) $$ P(2,0,0), \quad Q(0,2,-2), \quad R(0,0,4) $$

Find the component of \(\mathbf{u}\) along \(\mathbf{v}\) $$ \mathbf{u}=\langle- 3,5\rangle, \quad \mathbf{v}=\langle 1 / \sqrt{2}, 1 / \sqrt{2}\rangle $$

Find the area of \(\triangle P Q R\) $$ P(2,1,0), Q(0,0,-1), R(-4,2,0) $$

\(23-26\) Sketch representations of the given vector with initial points at \((0,0),(2,3),\) and \((-3,5)\). $$ \mathbf{u}=\langle 0,-9\rangle $$

Two vectors \(\mathbf{u}\) and \(\mathbf{v}\) are given. Find their dot product \(\mathbf{u} \cdot \mathbf{v} .\) $$ \mathbf{u}=6 \mathbf{i}-4 \mathbf{j}-2 \mathbf{k}, \quad \mathbf{v}=\frac{5}{6} \mathbf{i}+\frac{3}{2} \mathbf{j}-\mathbf{k} $$

A description of a line is given. Find parametric equations for the line. The line crosses the \(z\) -axis where \(z=4\) and crosses the \(x y-\) plane where \(x=2\) and \(y=5 .\)

Find the component of \(\mathbf{u}\) along \(\mathbf{v}\) $$ \mathbf{u}=7 \mathbf{i}-24 \mathbf{j}, \quad \mathbf{v}=\mathbf{j} $$

Find the area of \(\triangle P Q R\) $$ P(6,0,0), Q(0,-6,0), R(0,0,-6) $$

\(27-30\) Write the given vector in terms of \(\mathbf{i}\) and \(\mathbf{j}\) . $$ \mathbf{u}=\langle 1,4\rangle $$

Two vectors \(\mathbf{u}\) and \(\mathbf{v}\) are given. Find their dot product \(\mathbf{u} \cdot \mathbf{v} .\) $$ \mathbf{u}=3 \mathbf{j}-2 \mathbf{k}, \quad \mathbf{v}=\frac{5}{6} \mathbf{i}-\frac{5}{3} \mathbf{j} $$

A description of a line is given. Find parametric equations for the line. The line crosses the \(x\) -axis where \(x=-2\) and crosses the \(z\) -axis where \(z=10\) .

Find the component of \(\mathbf{u}\) along \(\mathbf{v}\) $$ \mathbf{u}=7 \mathbf{i}, \quad \mathbf{v}=8 \mathbf{i}+6 \mathbf{j} $$

Find the area of \(\triangle P Q R\) $$ P(3,-2,6), Q(-1,-4,-6), R(3,4,6) $$

\(27-30\) Write the given vector in terms of \(\mathbf{i}\) and \(\mathbf{j}\) . $$ \mathbf{u}=\langle- 2,10\rangle $$

Determine whether or not the given vectors are perpendicular. $$ \langle 4,-2,-4\rangle,\langle 1,-2,2\rangle $$

A description of a line is given. Find parametric equations for the line. The line perpendicular to the \(x z\) -plane that contains the point \((2,-1,5) .\)

(a) Calculate proj, \(\mathbf{u}\) . (b) Resolve \(\mathbf{u}\) into \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2},\) where \(\mathbf{u}_{1}\) is parallel to \(\mathbf{v}\) and \(\mathbf{u}_{2}\) is orthogonal to \(\mathbf{v} .\) $$ \mathbf{u}=\langle- 2,4\rangle, \quad \mathbf{v}=\langle 1,1\rangle $$

Three vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) are given. (a) Find their scalar triple product \(\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c}) .\) (b) Are the vectors coplanar? If not, find the volume of the parallelepiped that they determine. $$ \mathbf{a}=\langle 1,2,3\rangle, \quad \mathbf{b}=\langle- 3,2,1\rangle, \quad \mathbf{c}=\langle 0,8,10\rangle $$

\(27-30\) Write the given vector in terms of \(\mathbf{i}\) and \(\mathbf{j}\) . $$ \mathbf{u}=\langle 3,0\rangle $$

Determine whether or not the given vectors are perpendicular. $$ 4 \mathbf{j}-\mathbf{k}, \quad \mathbf{i}+2 \mathbf{j}+9 \mathbf{k} $$