Chapter 9

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

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

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 _________.

The angle \(\theta\) between the vectors \(\mathbf{u}\) and \(\mathbf{v}\) satisfies \(\cos \theta=\) So if \(\mathbf{u}\) and \(\mathbf{v}\) are perpendicular, then \(u \cdot v=\) ________ If \(\mathbf{u}=\langle 4,5,6\rangle\) and \(\mathbf{v}=\langle 3,0,-2\rangle\) then \(\mathbf{u} \cdot \mathbf{v}=\) _________ so u and v are _____________

Let \(a=\left\langle a_{1}, a_{2}\right\rangle\) 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 _____.

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

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 \(\mathbf{v}\) with initial point \(P\) and terminal point \(Q .\) $$P(1,-1,0), Q(0,-2,5)$$

For the given vectors \(\mathbf{a}\) and \(\mathbf{b},\) 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 \(\mathbf{v}\). $$P(0,-5,3), \quad \mathbf{v}=\langle 2,0,-4\rangle$$

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 \(\mathbf{v}\) with initial point \(P\) and terminal point \(Q .\) $$P(1,2,-1), Q(3,-1,2)$$

For the given vectors \(\mathbf{a}\) and \(\mathbf{b},\) 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 \(\mathbf{v}\). $$P(3,2,1), \quad \mathbf{v}=\langle 0,-4,2\rangle$$

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

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

For the given vectors \(\mathbf{a}\) and \(\mathbf{b},\) 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 \(\mathbf{v}\). $$P(0,0,0), \quad \mathbf{v}=\langle- 4,3,5\rangle$$

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

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

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

For the given vectors \(\mathbf{a}\) and \(\mathbf{b},\) 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 \(\mathbf{v}\). $$P(1,0,-2), \quad \mathbf{v}=2 \mathbf{i}-5 \mathbf{k}$$

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

Describe and sketch the surface represented by the given equation. $$x=4$$

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

For the given vectors \(\mathbf{a}\) and \(\mathbf{b},\) 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 point \(P\) and is parallel to the vector \(\mathbf{v}\). $$P(1,1,1), \quad \mathbf{v}=\mathbf{i}-\mathbf{j}+\mathbf{k}$$

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

Describe and sketch the surface represented by the given equation. $$y=-2$$

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

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 \(Q\) $$P(1,-3,2), \quad Q(2,1,-1)$$

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

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

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

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

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

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

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

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

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

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

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. $$\mathbf{a}=3 \mathbf{j}+5 \mathbf{k}, \quad \mathbf{b}=-\mathbf{i}+2 \mathbf{k}$$

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