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

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

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 form 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 objects traveling on the lines $$\vec{P}_1(s) = \langle 2+3s, 4+s, 3+5s \rangle$$ and $$\vec{P}_2(t) = \langle 5+2t, 3-5t, -5+10t \rangle$$, do their paths intersect?

Accepted Answer

Yes, at $$s=0$$, $$t=1$$

Question 12

If $$\vec{u} = \langle 1, 2, 3 angle$$ and $$\vec{v} = \langle -1, -3, 0 angle$$, what is $$\vec{u} 	imes \vec{v}$$?

Accepted Answer

Question 13

Which of the following statements is true about skew lines in $$\mathbb{R}^3$$?

Accepted Answer

They do not intersect and are not parallel

Question 14

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

Accepted Answer

Their normal vectors are orthogonal

Question 15

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 16

A vector with length 3 in the same direction as $$\vec{v} is $$given by:

Accepted Answer

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

Vectors, Lines, and Planes in Multivariable Calculus

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