Find a unit vector that has:

(a) The same direction as $\textbf{a}$; and

(b) A direction opposite to that of $\textbf{a}$.

$$\textbf{a}=\left<1,2,2\right>$$

Write the given equation:

(a) In cylindrical coordinates; and

(b) In spherical coordinates.

$$2x+3y-4z=12$$

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that shows it is false.

The surface $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=4$ is an ellipsoid obtained by stretching the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ by a factor of $2$ in each of the $x$-, $y$-, and $z$-directions.

Show that the distance $D$ between the parallel planes $ax+by+cz=d_1$ and $ax+by+cz=d_2$ is

$$D=\frac{\lvert a_1-d_2\rvert}{\sqrt{a^2+b^2+c^2}}$$

Write the given equation:

(a) In cylindrical coordinates; and

(b) In spherical coordinates.

$$x^2+y^2=9$$

Determine whether $\textbf{b}$ is parallel to $\textbf{a}=\textbf{i}-2\textbf{j}+5\textbf{k}$.

$$\textbf{b}=\left<-2,4,-10\right>$$

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that shows it is false.

The graph of $9x^2+4z^2=1$ and $y=0$ is the graph of an ellipse lying in the xz-plane.

Refer to the previous exercise. Let $\textbf{a}=\left<2,1\right>$ and $\textbf{b}=\left<4,5\right>$.

a. Find $\textbf{b}-\text{proj}_\textbf{a}\textbf{b}$.

b. Sketch the vectors $\textbf{a},\ \textbf{b},\ \text{proj}_\textbf{a}\textbf{b}$ and $\textbf{b}-\text{proj}_\textbf{a}\textbf{b}$.

c. Show that $\textbf{b}-\text{proj}_\textbf{a}\textbf{b}$ is orthogonal to $\textbf{a}$.

Find the distance between the point $(1,4,2)$ and the line passing through the points $(-1, 3, -1)$ and $(1, 2, 3)$.

Identify the surface whose equation is given.

$$z=-2$$

Find a vector $\textbf{a}$ that has the given length and makes an angle $\theta$ with the positive $x$-axis.

$$\lvert \textbf{a}\rvert =1; \hspace{6pt}\theta=\frac{\pi}{2}$$

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example to show that it is false.

If $\mathbf{c}$ is a constant vector, then $\frac{d}{dt}(\mathbf{c})=0$.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that shows it is false.

If $\mathbf{a} \neq 0,\ \mathbf{a}\cdot \mathbf{b}=\mathbf{a}\cdot \mathbf{c},$ and $\mathbf{a}\times \mathbf{b}=\mathbf{a}\times \mathbf{c}$ then $\mathbf{b}=\mathbf{c}$.

Determine whether $\textbf{b}$ is parallel to $\textbf{a}=\textbf{i}-2\textbf{j}+5\textbf{k}$.

$$\textbf{b}=2\textbf{i}-3\textbf{j}+10\textbf{k}$$