Q: Let $$\vec{a} = \langle 2, -1, 3 \rangle$$ and $$\vec{b} = \langle 1, 4, -2 \rangle. $$What is the angle $$\theta$$ between $$\vec{a}$$ and $$\vec{b}$$? (Give your answer in radians.)

$$\theta = \arccos\left(\dfrac{2(1) + (-1)(4) + 3(-2)}{\sqrt{2^2$ + (-1)^2$ + 3^2$} \cdot \sqrt{1^2$ + 4^2$ + (-2)^2$}}\right)$$

Question 1

Let $$\vec{a} = \langle 2, -1, 3 angle$$ and $$\vec{b} = \langle 1, 4, -2 angle. $$What is the angle $$	heta$$ between $$\vec{a}$$ and $$\vec{b}$$? (Give your answer in radians.)

Accepted Answer

$$	heta = \arccos\left(\dfrac{2(1) + (-1)(4) + 3(-2)}{\sqrt{2^2$ + (-1)^2$ + 3^2$} \cdot \sqrt{1^2$ + 4^2$ + (-2)^2$}}ight)$$

Question 2

Given vectors $$\vec{F} = \langle 3, 4 \rangle$$ and $$\vec{v} = \langle 1, 0 \rangle$$, what is the projection of $$\vec{F}$$ onto $$\vec{v}$$?

Accepted Answer

$$\operatorname{proj}_{\vec{v}} \vec{F} = \left(\dfrac{3}{1^2$}\right) \langle 1, 0 \rangle = \langle 3, 0 \rangle$$

Question 3

Which of the following is the correct formula for the cross product $$\vec{a} 	imes \vec{b}$$ for $$\vec{a} = \langle a_1$, a_2$, a_3$ angle$$ and $$\vec{b} = \langle b_1$, b_2$, b_3$ angle$$?

Accepted Answer

$$\langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle$$

Question 4

Find the equation of the plane passing through the point $$(1,2,3)$$ and perpendicular to the vector $$\vec{n} = \langle 2, -1, 4 \rangle.$$

Accepted Answer

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

Question 5

Given $$\vec{r}(t) = \langle 3t, 4t, 0 \rangle$$, find the speed $$|\vec{v}(t)| at $$any time $$t.$$

Accepted Answer

$$|\vec{v}(t)| = \sqrt{3^2$ + 4^2$} = 5$$

Question 6

A projectile is launched from the ground with initial velocity $$\vec{v}_0 = \langle 10, 20 \rangle m/s. $$Neglecting air resistance and using $$g = 9.8 m/s$^2$$, what is the position vector $$\vec{r}(t) at $$time $$t$$?

Accepted Answer

$$\vec{r}(t) = \langle 10t, 20t - 4.9t^2$ \rangle$$

Question 7

If $$|\vec{r}(t)| is $$constant for all $$t$$, what is the relationship between $$\vec{r}(t)$$ and $$\vec{r}'(t)$$?

Accepted Answer

$$\vec{r}(t)$$ and $$\vec{r}'(t)$$ are orthogonal

Question 8

Given $$\vec{r}(t) = \langle \cos t, \sin t, 0 \rangle$$, find the curvature $$\kappa at $$any $$t.$$

Accepted Answer

$$\kappa = 1$$

Question 9

Find the distance from the point $$(1,2,3) to $$the plane $$2x - y + 2z = 5.$$

Accepted Answer

$$\dfrac{|2(1) - 1(2) + 2(3) - 5|}{\sqrt{2^2$ + (-1)^2$ + 2^2$}}$$

Question 10

Given $$\vec{r}(t) = \langle t, t^2$, t^3$ \rangle$$, find $$\vec{v}(t)$$ and $$\vec{a}(t).$$

Accepted Answer

$$\vec{v}(t) = \langle 1, 2t, 3t^2$ \rangle$$, $$\vec{a}(t) = \langle 0, 2, 6t \rangle$$

Question 11

Which of the following is the correct formula for the arc length $$s of a $$curve $$\vec{r}(t)$$ from $$t=a to$$ $$t=b$$?

Accepted Answer

$$s = \int_a^b |\vec{r}'(t)| \, dt$$

Question 12

Given $$\vec{a} = \langle 1, 2, 3 angle$$ and $$\vec{b} = \langle 4, 5, 6 angle$$, what is the value of the box product $$\vec{a} \cdot (\vec{b} 	imes \vec{c})$$ for $$\vec{c} = \langle 7, 8, 9 angle$$?

Accepted Answer

$$0$$

Question 13

Which of the following statements about the cross product is TRUE?

Accepted Answer

The cross product of two vectors is orthogonal to both vectors.

Question 14

Given the line $$\vec{r}(t) = \langle 1, 2, 3 \rangle + t \langle 2, -1, 4 \rangle$$, which point lies on the line?

Accepted Answer

$$(3, 1, 7)$$

Question 15

If $$\vec{T} is $$the unit tangent vector to a curve in $$\mathbb{R}^2$$, which of the following is a shortcut to find the unit normal vector $$\vec{N}$$?

Accepted Answer

$$\vec{N} = \langle -T_2$, T_1$ \rangle$$

Question 16

Given $$\vec{a} = \langle 2, 0, 0 angle$$ and $$\vec{b} = \langle 0, 3, 0 angle$$, what is $$\vec{a} 	imes \vec{b}$$?

Accepted Answer

$$\langle 0, 0, 6 \rangle$$

Question 17

A particle moves along $$\vec{r}(t) = \langle t, t^2$ \rangle. $$What is the acceleration vector $$\vec{a}(t)$$?

Accepted Answer

$$\langle 0, 2 \rangle$$

Question 18

Which of the following expresses the acceleration $$\vec{a} in $$terms of the unit tangent $$\vec{T}$$ and unit normal $$\vec{N}$$ vectors?

Accepted Answer

$$\vec{a} = a_T \vec{T} + a_N \vec{N}$$

Question 19

Given $$\vec{r}(t) = \langle t, t^2$, 0 \rangle$$, what is the formula for the curvature $$\kappa at$$ $$t$$?

Accepted Answer

$$\kappa = \dfrac{2}{\left(1 + 4t^2$\$$right)^{3/2}$$}$$

Calculus III Study Guide: Vectors, Geometry of Space, and Vector-Valued Functions (Chapters 12 & 13)

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