Question 1

Given the parametric equations $$x = 2 \cos t$$, $$y = 2 \sin t$$, for $$0 \leq t \leq 2\pi$$, what is the length of the curve traced out by these equations?

Accepted Answer

$$4\pi$$

Question 2

Convert the polar coordinates $$(2, \frac{\pi}{4}) to $$Cartesian coordinates $$(x, y). $$Which of the following is correct?

Accepted Answer

$$(\sqrt{2}, \sqrt{2})$$

Question 3

Which of the following is the correct equation for the graph of a circle centered at the origin with radius $$3 in $$polar coordinates?

Accepted Answer

$$r = 3$$

Question 4

Given the vector-valued function $$\vec{r}(t) = \langle t, t^2$, t^3$ \rangle$$, what is $$\vec{r}'(t)$$?

Accepted Answer

$$\langle 1, 2t, 3t^2$ \rangle$$

Question 5

Find the equation of the plane passing through the point $$(1, 2, 3)$$ and normal to the vector $$\langle 2, -1, 4 \rangle.$$

Accepted Answer

$$2(x-1) - 1(y-2) + 4(z-3) = 0$$

Question 6

Calculate the distance from the point $$(5, 4, -2) to $$the plane $$2x - y + 2z = 7.$$

Accepted Answer

Question 7

Which of the following is the vector equation of the line passing through the point $$(1, 2, 3)$$ and parallel to the vector $$\langle 4, -2, 5 \rangle$$?

Accepted Answer

$$\vec{r}(t) = \langle 1, 2, 3 \rangle + t\langle 4, -2, 5 \rangle$$

Question 8

Given the points $$A(1, 2, 3)$$ and $$B(4, 6, 8)$$, what is the vector from $$A to$$ $$B$$?

Accepted Answer

$$\langle 3, 4, 5 \rangle$$

Question 9

Find the intersection point of the line $$x = 1 + 2t$$, $$y = 2 - t$$, $$z = 3t$$ and the plane $$x + y + z = 7.$$

Accepted Answer

Question 10

Which of the following is the correct parametric equation for the line passing through $$(1, 2, 3)$$ and $$(4, 5, 6)$$?

Accepted Answer

$$x = 1 + 3t,\ y = 2 + 3t,\ z = 3 + 3t$$

Question 11

Given the vector $$\vec{a} = \langle 2, -1, 3 \rangle$$ and $$\vec{b} = \langle 1, 4, -2 \rangle$$, what is $$\vec{a} \cdot \vec{b}$$?

Accepted Answer

Question 12

Which of the following is the correct equation for a plane passing through $$(1, 2, 3)$$ and perpendicular to the vector $$\langle 2, -1, 4 \rangle$$?

Accepted Answer

$$2(x-1) - 1(y-2) + 4(z-3) = 0$$

Question 13

If $$\vec{a} = \langle 1, 2, 3 angle$$ and $$\vec{b} = \langle 4, 5, 6 angle$$, what is $$\vec{a} 	imes \vec{b}$$?

Accepted Answer

$$\langle -3, 6, -3 \rangle$$

Question 14

Which of the following is the correct formula for the distance from a point $$(x_0$, y_0$, z_0$) to $$the plane $$Ax + By + Cz + D = 0$$?

Accepted Answer

$$\displaystyle \frac{|Ax_0$ + By_0$ + Cz_0$ + D|}{\sqrt{A^2$ + B^2$ + C^2$}}$$

Question 15

Given the parametric equations $$x = 3t + 1$$, $$y = 2t - 1$$, $$z = t$$, what is the direction vector of the line?

Accepted Answer

$$\langle 3, 2, 1 \rangle$$

Question 16

Which of the following is the correct equation for a sphere of radius $$r$$ centered at $$(a, b, c)$$?

Accepted Answer

$$(x-a)^2$$ + $$(y-b)^2$$ + $$(z-c)^2$$ = $$r^2$$

Question 17

If a particle moves along the curve $$\vec{r}(t) = \langle \cos t, \sin t, t \rangle$$, what is the speed of the particle at $$t = 0$$?

Accepted Answer

$$\sqrt{2}$$

Question 18

Given the plane $$2x - y + 2z = 7$$, which of the following points lies on the plane?

Accepted Answer

$$(1, 1, 2)$$

Question 19

Which of the following is the correct formula for the dot product of two vectors $$\vec{a} = \langle a_1$, a_2$, a_3$ \rangle$$ and $$\vec{b} = \langle b_1$, b_2$, b_3$ \rangle$$?

Accepted Answer

$$a_1b_1 + a_2b_2 + a_3b_3$$

Study Notes: Calculus and Analytic Geometry – Polar Coordinates, Equations of Lines and Planes, and Distance in Space

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