Question 1

Which of the following best describes the set of points in space whose coordinates satisfy the equations $$y = 0$$ and $$z = 0$$?

Accepted Answer

The $$x-$$axis

Question 2

Which equation describes the circle of radius $$1$$ centered at $$(0,8,0) in $$the $$yz-$$plane?

Accepted Answer

$$(y-8)^2$$ + $$z^2 = 1$, x = 0$

Question 3

Let $$\mathbf{u} = \langle 4, -1 \rangle. $$What is the magnitude of $$2\mathbf{u}$$?

Accepted Answer

$$2\sqrt{17}$$

Question 4

Given points $$A(-3, -7, 4)$$ and $$B(10, 7, 4)$$, what is the vector $$\overrightarrow{AB} in $$component form?

Accepted Answer

$$13\mathbf{i} + 14\mathbf{j} + 0\mathbf{k}$$

Question 5

Let $$\mathbf{v} = 4\mathbf{i} - 3\mathbf{k}$$ and $$\mathbf{u} = \mathbf{i} + \mathbf{j} + \mathbf{k}. $$What is the dot product $$\mathbf{v} \cdot \mathbf{u}$$?

Accepted Answer

$$4$$

Question 6

Find the cosine of the angle $$	heta$$ between $$\mathbf{v} = 4\mathbf{i} - 3\mathbf{k}$$ and $$\mathbf{u} = \mathbf{i} + \mathbf{j} + \mathbf{k}.$$

Accepted Answer

$$\dfrac{4}{\sqrt{26}\sqrt{3}}$$

Question 7

Given $$\mathbf{u} = \sqrt{3}\mathbf{i} - 5\mathbf{j}$$ and $$\mathbf{v} = \sqrt{3}\mathbf{i} + \mathbf{j} - 2\mathbf{k}$$, what is the angle between $$\mathbf{u}$$ and $$\mathbf{v} in $$radians?

Accepted Answer

$$\arccos\left(\dfrac{-2}{\sqrt{28}\sqrt{7}}\right)$$

Question 8

What is the area of the parallelogram with vertices $$A(-1,2)$$, $$B(2,0)$$, $$C(9,2)$$, and $$D(6,4)$$?

Accepted Answer

$$20$$

Question 9

Find the equation of the plane passing through the point $$P_0 = (0$$, 3, -2)$$ with normal vector $$\mathbf{n} = 2\mathbf{i} - 8\mathbf{j} - \mathbf{k}$$.

Accepted Answer

$$2x - 8y - z = -22$$

Question 10

What is the distance from the point $$P(6, -8, -3) to $$the line $$x = 6 + 3t$$, $$y = -8 + 9t$$, $$z = -3$$?

Accepted Answer

$$0$$

Question 11

Evaluate the limit: 
$$
\lim_{t 	o \pi} \left( \sin t / 3 \; \mathbf{i} + \cos(3/4\; t) \; \mathbf{j} + 	an(2/3\; t) \; \mathbf{k} ight)
$$

Accepted Answer

$$0\mathbf{i} -1\mathbf{j} + 0\mathbf{k}$$

Question 12

Given the path $$\mathbf{r}(t) = (t - \sin t)\mathbf{i} + (1 - \cos t)\mathbf{j}$$, what is the velocity vector at $$t = \pi$$?

Accepted Answer

$$\mathbf{i}$$

Question 13

Evaluate the integral $$\$$int_0$$^1 \left[ 7t^3$\mathbf{i} + 8\mathbf{j} + (t+6)\mathbf{k} \right] dt.$$

Accepted Answer

$$\left(\dfrac{7}{4}\right)\mathbf{i} + 8\mathbf{j} + \left(6.5\right)\mathbf{k}$$

Question 14

Solve the initial value problem: $$\dfrac{d\mathbf{r}}{dt} = 6(t+1$$)^{1/2}$$\mathbf{i} + 5e$$^{-t}$$\mathbf{j} + \dfrac{1}{t+1}\mathbf{k}$$, with $$\mathbf{r}(0) = \mathbf{k}. $$What is $$\mathbf{r}(t)$$?

Accepted Answer

$$4(t+1)^{3/2}$$\mathbf{i} - $$5e^{-t}$$\mathbf{j} + (\ln(t+1) + 1)\mathbf{k}$$

Question 15

Given $$\mathbf{r}(t) = (3\cos t)\mathbf{i} + (3\sin t)\mathbf{j} + (\sqrt{3}t)\mathbf{k}$$, what is the unit tangent vector $$\mathbf{T}(t)$$?

Accepted Answer

$$\dfrac{-\sin t\; \mathbf{i} + \cos t\; \mathbf{j} + (1/\sqrt{3})\mathbf{k}}{2}$$

Question 16

Find the arc length of the curve $$\mathbf{r}(t) = 4\cos t\; \mathbf{i} + 4\sin t\; \mathbf{j} + 7t\; \mathbf{k}$$ for $$0 \leq t \leq \pi.$$

Accepted Answer

$$\$$int_0$$^{\pi} \sqrt{16 + 49} dt = \sqrt{65}\pi$$

Question 17

Which of the following is the correct formula for the projection of $$\mathbf{u}$$ onto $$\mathbf{v}$$?

Accepted Answer

$$\mathrm{proj}_{\mathbf{v}} \mathbf{u} = \dfrac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|^2} \mathbf{v}$$

Question 18

If $$\mathbf{u} = 5\mathbf{i} + 6\mathbf{j} - 4\mathbf{k}$$ and $$\mathbf{v} = \mathbf{0}$$, what is $$\mathbf{u} 	imes \mathbf{v}$$?

Accepted Answer

$$\mathbf{0}$$

Math 275 Exam 1 Review: Calculus Guidance (Vectors, Planes, Limits, Integrals, Arc Length)

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