Chapter 10

Let \(\mathbf{a}=\left\langle a_{1}, a_{2}\right\rangle\) and \(\mathbf{b}=\left\langle b_{1}, b_{2}\right\rangle\) be nonzero vectors in the plane, and let \(\theta\) be the angle between them. The dot product of a and b is defined by \(\mathbf{a} \cdot \mathbf{b}=\) ______ The dot product of two vectors is a ______, not a vector.

The cross product of the vectors \(\mathbf{a}=\left\langle a_{1}, a_{2}, a_{3}\right\rangle\) and \(\mathbf{b}=\left\langle b_{1}, b_{2}, b_{3}\right\\}\) is the vector $$ \mathbf{a} \times \mathbf{b}=\left|\begin{array}{lll}{\mathbf{i}} & {\mathbf{j}} & {\mathbf{k}} \\ {} & {} & {}\end{array}\right| $$ = _____ \(\mathbf{i}+\) _____ \(\mathbf{j}+\) _____ \(\mathbf{k}\) So the cross product of \(\mathbf{a}=\langle 1,0,1\rangle\) and \(\mathbf{b}=\langle 2,3,0\rangle\) is a \(\times \mathbf{b}=\) _____.

The plane containing the point \(P\left(x_{0}, y_{0}, z_{0}\right)\) and having the normal vector \(\mathbf{n}=\langle a, b, c\rangle\) is described algebraically by the equation ______

Let \(\mathbf{a}=\left\langle a_{1}, a_{2}\right\rangle\) and \(\mathbf{b}=\left\langle b_{1}, b_{2}\right\rangle\) be nonzero vectors in the plane, and let \(\theta\) be the angle between them. The angle \(\theta\) satisfies \(\cos \theta=\) _____ So if \(\mathbf{a} \cdot \mathbf{b}=0,\) the vectors are _____

The cross product of two vectors a and b is _____ to a and to b. Thus if both vectors a and b lie in a plane, the vector \(\mathbf{a} \times \mathbf{b}\) is _____ to the plane.

Find parametric equations for the line that passes through the point \(P\) and is parallel to the vector \(\mathbf{v} .\) $$ P(1,0,-2), \quad \mathbf{v}=\langle 3,2,-3\rangle $$

\(3-6\) Two points \(P\) and \(Q\) are given. (a) Plot \(P\) and \(Q .(b)\) Find the distance between \(P\) and \(Q\) $$ P(3,1,0), Q(-1,2,-5) $$

Find the vector v with initial point \(P\) and terminal point \(Q\) . $$ P(1,-1,0), Q(0,-2,5) $$

For the given vectors a and b, find the cross product \(\mathbf{a} \times \mathbf{b}\). $$ \mathbf{a}=\langle 1,0,-3\rangle, \quad \mathbf{b}=\langle 2,3,0\rangle $$

Find parametric equations for the line that passes through the point \(P\) and is parallel to the vector \(\mathbf{v} .\) $$ P(0,-5,3), \quad \mathbf{v}=\langle 2,0,-4\rangle $$

\(3-6\) Two points \(P\) and \(Q\) are given. (a) Plot \(P\) and \(Q .(b)\) Find the distance between \(P\) and \(Q\) $$ P(5,0,10), Q(3,-6,7) $$

Find the vector v with initial point \(P\) and terminal point \(Q\) . $$ P(1,2,-1), Q(3,-1,2) $$

For the given vectors a and b, find the cross product \(\mathbf{a} \times \mathbf{b}\). $$ \mathbf{a}=\langle 0,-4,1\rangle, \quad \mathbf{b}=\langle 1,1,-2\rangle $$

Find parametric equations for the line that passes through the point \(P\) and is parallel to the vector \(\mathbf{v} .\) $$ P(3,2,1), \quad \mathbf{v}=\langle 0,-4,2\rangle $$

Find (a) \(\mathbf{u} \cdot \mathbf{v}\) and (b) the angle between \(\mathbf{u}\) and \(\mathbf{v}\) to the nearest degree. $$ \mathbf{u}=\langle 2,0\rangle, \quad \mathbf{v}=\langle 1,1\rangle $$

\(3-6\) Two points \(P\) and \(Q\) are given. (a) Plot \(P\) and \(Q .(b)\) Find the distance between \(P\) and \(Q\) $$ P(-2,-1,0), Q(-12,3,0) $$

Find the vector v with initial point \(P\) and terminal point \(Q\) . \(P(6,-1,0), Q(0,-3,0)\)

For the given vectors a and b, find the cross product \(\mathbf{a} \times \mathbf{b}\). $$ \mathbf{a}=\langle 6,-2,8\rangle, \quad \mathbf{b}=\langle- 9,3,-12\rangle $$

Find parametric equations for the line that passes through the point \(P\) and is parallel to the vector \(\mathbf{v} .\) $$ P(0,0,0), \quad \mathbf{v}=\langle- 4,3,5\rangle $$

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

Find the vector v with initial point \(P\) and terminal point \(Q\) . $$ P(1,-1,-1), Q(0,0,-1) $$

\(3-6\) Two points \(P\) and \(Q\) are given. (a) Plot \(P\) and \(Q .(b)\) Find the distance between \(P\) and \(Q\) $$ P(5,-4,-6), Q(8,-7,4) $$

For the given vectors a and b, find the cross product \(\mathbf{a} \times \mathbf{b}\). $$ \mathbf{a}=\langle- 2,3,4\rangle, \quad \mathbf{b}=\left\langle\frac{1}{6},-\frac{1}{4},-\frac{1}{3}\right\rangle $$

Find parametric equations for the line that passes through the point \(P\) and is parallel to the vector \(\mathbf{v} .\) $$ P(1,0,-2), \quad \mathbf{v}=2 \mathbf{i}-5 \mathbf{k} $$

Find (a) \(\mathbf{u} \cdot \mathbf{v}\) and (b) the angle between \(\mathbf{u}\) and \(\mathbf{v}\) to the nearest degree. $$ \mathbf{u}=\langle 2,7\rangle, \quad \mathbf{v}=\langle 3,1\rangle $$

If the vector \(\mathbf{v}\) has initial point \(P,\) what is its terminal point? $$ \mathbf{v}=\langle 3,4,-2\rangle, P(2,0,1) $$

\(7-10\) . Describe and sketch the surface represented by the given equation. \(x=4\)

For the given vectors a and b, find the cross product \(\mathbf{a} \times \mathbf{b}\). $$ \mathbf{a}=\mathbf{i}+\mathbf{j}+\mathbf{k}, \quad \mathbf{b}=3 \mathbf{i}-4 \mathbf{k} $$

Find parametric equations for the line that passes through the point \(P\) and is parallel to the vector \(\mathbf{v} .\) $$ P(1,1,1), \quad \mathbf{v}=\mathbf{i}-\mathbf{j}+\mathbf{k} $$

Find (a) \(\mathbf{u} \cdot \mathbf{v}\) and (b) the angle between \(\mathbf{u}\) and \(\mathbf{v}\) to the nearest degree. $$ \mathbf{u}=\langle- 6,6\rangle, \quad \mathbf{v}=\langle 1,-1\rangle $$

If the vector \(\mathbf{v}\) has initial point \(P,\) what is its terminal point? $$ \mathbf{v}=\langle 0,0,1\rangle, P(0,1,-1) $$

\(7-10\) . Describe and sketch the surface represented by the given equation. \(y=-2\)

For the given vectors a and b, find the cross product \(\mathbf{a} \times \mathbf{b}\). $$ \mathbf{a}=3 \mathbf{i}-\mathbf{j}, \quad \mathbf{b}=-3 \mathbf{j}+\mathbf{k} $$

Find parametric equations for the line that passes through the points \(P\) and \(Q .\) $$ P(1,-3,2), \quad Q(2,1,-1) $$

Find (a) \(\mathbf{u} \cdot \mathbf{v}\) and (b) the angle between \(\mathbf{u}\) and \(\mathbf{v}\) to the nearest degree. $$ \mathbf{u}=\langle 3,-2\rangle, \quad \mathbf{v}=\langle 1,2\rangle $$

If the vector \(\mathbf{v}\) has initial point \(P,\) what is its terminal point? $$ \mathbf{v}=\langle- 2,0,2\rangle, P(3,0,-3) $$

\(7-10\) . Describe and sketch the surface represented by the given equation. \(z=8\)

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. $$ \mathbf{a}=\langle 1,1,-1\rangle, \quad \mathbf{b}=\langle- 1,1,-1\rangle $$

Find parametric equations for the line that passes through the points \(P\) and \(Q .\) $$ P(2,-1,-2), \quad Q(0,1,-3) $$

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

If the vector \(\mathbf{v}\) has initial point \(P,\) what is its terminal point? $$ \mathbf{v}=\langle 23,-5,12\rangle, \quad P(-6,4,2) $$

\(7-10\) . Describe and sketch the surface represented by the given equation. \(y=-1\)

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. $$ \mathbf{a}=\langle 2,5,3\rangle, \quad \mathbf{b}=\langle 3,-2,-1\rangle $$

Find parametric equations for the line that passes through the points \(P\) and \(Q .\) $$ P(1,1,0), \quad Q(0,2,2) $$

Find (a) \(\mathbf{u} \cdot \mathbf{v}\) and (b) the angle between \(\mathbf{u}\) and \(\mathbf{v}\) to the nearest degree. $$ \mathbf{u}=-5 \mathbf{j}, \quad \mathbf{v}=-\mathbf{i}-\sqrt{3} \mathbf{j} $$

Find the magnitude of the given vector. $$ \langle- 2,1,2\rangle $$

\(11-14\) . Find an equation of a sphere with the given radius \(r\) and center \(C .\) $$ r=5 ; \quad C(2,-5,3) $$

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. $$ \mathbf{a}=\frac{1}{2} \mathbf{i}-\mathbf{j}+\frac{2}{3} \mathbf{k}, \quad \mathbf{b}=6 \mathbf{i}-12 \mathbf{j}-6 \mathbf{k} $$

Find parametric equations for the line that passes through the points \(P\) and \(Q .\) $$ P(3,3,3), \quad Q(7,0,0) $$

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