Question 1

Given points $$P(1,2)$$ and $$Q(3,-5)$$, what is the vector $$\overrightarrow{PQ}$$?

Accepted Answer

$$\langle 2, -7 \rangle$$

Question 2

What is the magnitude of the vector $$\vec{u} = \langle 1, 2 \rangle$$?

Accepted Answer

$$\sqrt{5}$$

Question 3

Which of the following is a unit vector in the same direction as $$\vec{v} = \langle 1, 2, 3 \rangle$$?

Accepted Answer

$$\left\langle \frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}} \right\rangle$$

Question 4

What is the result of $$\vec{v} + \vec{w} if$$ $$\vec{v} = \langle 1, 2, 3 \rangle$$ and $$\vec{w} = \langle 4, 5, 6 \rangle$$?

Accepted Answer

$$\langle 5, 7, 9 \rangle$$

Question 5

Which equation represents a sphere with center $$(h, k, d)$$ and radius $$r$$?

Accepted Answer

$$(x-h)^2$$ + $$(y-k)^2$$ + $$(z-d)^2$$ = $$r^2$$

Question 6

Calculate the cross product $$\vec{u} 	imes \vec{v}$$ for $$\vec{u} = \langle 2, 1, 1 angle$$ and $$\vec{v} = \langle 5, 0, 1 angle.$$

Accepted Answer

$$\langle 1, 3, -5 \rangle$$

Question 7

Which of the following is the vector equation of a line passing through $$(x_0$, y_0$, z_0$)$$ with direction vector $$\vec{v} = \langle a, b, c \rangle$$?

Accepted Answer

$$\vec{P}(t) = \langle x_0$, y_0$, z_0$ \rangle + t \langle a, b, c \rangle$$

Question 8

Given the line through $$P(0,1,2)$$ and $$Q(-3,4,7)$$, what is the parametric equation for $$x(t)$$?

Accepted Answer

$$x(t) = -3t$$

Question 9

Which of the following is the equation of a plane with normal vector $$\vec{n} = \langle a, b, c \rangle$$ passing through $$(x_0$, y_0$, z_0$)$$?

Accepted Answer

$$a(x-x_0$) + b(y-y_0$) + c(z-z_0$) = 0$$

Question 10

If two planes have normal vectors $$\vec{n}_1$$ and $$\vec{n}_2$$, when are the planes parallel?

Accepted Answer

When $$\vec{n}_1$$ and $$\vec{n}_2$$ are parallel

Question 11

Given two lines in $$\mathbb{R}^3$$ with direction vectors $$\vec{u} = \langle 1, 0, 1 \rangle$$ and $$\vec{v} = \langle 1, 2, 3 \rangle$$, what is true about these lines?

Accepted Answer

They are skew lines and do not intersect

Question 12

If two planes are orthogonal, what is true about their normal vectors?

Accepted Answer

Their normal vectors are orthogonal

Question 13

What is the direction vector of the line of intersection of two planes with normal vectors $$\vec{n}_1$$ and $$\vec{n}_2$$?

Accepted Answer

$$\vec{n}_1 	imes \vec{n}_2$$

Question 14

Suppose two objects travel along the lines $$\vec{P}_1(s) = \langle 2+3s, 4+s, 3+5s \rangle$$ and $$\vec{P}_2(t) = \langle 5+2t, 3-5t, -5+10t \rangle. At $$what values of $$s$$ and $$t do $$their paths intersect?

Accepted Answer

$$s = 0, t = 1$$

Question 15

Which of the following best describes a unit vector?

Accepted Answer

A vector with magnitude 1

Question 16

If $$\vec{v} = \langle 1, 2, 3 \rangle$$, what is a vector of length 3 in the same direction as $$\vec{v}$$?

Accepted Answer

$$3 \frac{\vec{v}}{|\vec{v}|}$$

Question 17

Which of the following is the correct parametric form for a line through $$(x_0$, y_0$, z_0$)$$ with direction vector $$\langle a, b, c \rangle$$?

Accepted Answer

$$x(t) = x_0$ + at,\ y(t) = y_0$ + bt,\ z(t) = z_0$ + ct$$

Question 18

Given vectors $$\vec{u} = \langle 0, 1, 2 \rangle$$ and $$\vec{v} = \langle -1, -3, 0 \rangle$$, what is their cross product?

Accepted Answer

$$\langle 6, -2, 1 \rangle$$

Vectors, Lines, and Planes in Multivariable Calculus

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