Question 1

Given the points $$P = (-1, 1, 1)$$ and $$Q = (0, 3, -1)$$, what is the distance between $$P$$ and $$Q$$?

Accepted Answer

$$\sqrt{9}$$

Question 2

Which of the following equations represents the set of all points $$(x, y, z)$$ that are a distance $$3$$ from the point $$P = (-1, 1, 1)$$?

Accepted Answer

$$(x + 1)^2$ + (y - 1)^2$ + (z - 1)^2$ = 9$$

Question 3

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

Accepted Answer

$$3$$

Question 4

What is the cosine of the angle $$	heta$$ between $$\vec{a} = \langle 1, -2, 3 angle$$ and $$\vec{b} = \langle 2, 1, 1 angle$$?

Accepted Answer

$$\dfrac{3}{\sqrt{14} \sqrt{6}}$$

Question 5

What is the vector projection of $$\vec{b} = \langle 2, 1, 1 \rangle$$ onto $$\vec{a} = \langle 1, -2, 3 \rangle$$?

Accepted Answer

$$\left\langle \dfrac{3}{14}, -\dfrac{6}{14}, \dfrac{9}{14} \right\rangle$$

Question 6

Given $$\vec{a} = \langle 1, -2, 3 \rangle$$ and $$\vec{b} = \langle 2, 1, 1 \rangle$$, what is a vector orthogonal to both $$\vec{a}$$ and $$\vec{b}$$?

Accepted Answer

$$\langle -5, 5, 5 \rangle$$

Question 7

What is the area of the parallelogram formed by $$\vec{a} = \langle 1, -2, 3 \rangle$$ and $$\vec{b} = \langle 2, 1, 1 \rangle$$?

Accepted Answer

$$5\sqrt{3}$$

Question 8

Which of the following is the parametric equation for the line through points $$P = (1, 2, 3)$$ and $$Q = (0, 4, 4)$$?

Accepted Answer

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

Question 9

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

Accepted Answer

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

Question 10

If two lines in space have direction vectors $$\vec{v}_1 = \langle 1, 2, 3 \rangle$$ and $$\vec{v}_2 = \langle 2, 4, 6 \rangle$$, what is true about the lines?

Accepted Answer

The lines are parallel

Question 11

Given $$\vec{a} = \langle 1, -2, 3 angle$$ and $$\vec{b} = \langle 2, 1, 1 angle$$, what is $$|\vec{a} 	imes \vec{b}|$$?

Accepted Answer

$$5\sqrt{3}$$

Question 12

Which of the following statements is true about the dot product $$\vec{v} \cdot \vec{w}$$?

Accepted Answer

If $$\vec{v} \cdot \vec{w} = 0$$, then $$\vec{v}$$ and $$\vec{w}$$ are orthogonal

Question 13

Given the parametric equations $$x = 1 - t$$, $$y = 2 + 2t$$, $$z = 3 + t$$, what is the point on the line when $$t = 2$$?

Accepted Answer

$$( -1, 6, 5 )$$

Question 14

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

Accepted Answer

$$\langle -5, 5, 5 \rangle$$

Question 15

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

Accepted Answer

$$\left\langle \dfrac{1}{3}, \dfrac{2}{3}, \dfrac{2}{3} \right\rangle$$

Question 16

Given two lines in space with direction vectors $$\vec{v}_1 = \langle 1, 2, 3 \rangle$$ and $$\vec{v}_2 = \langle -2, -4, -6 \rangle$$, what is the relationship between the lines?

Accepted Answer

The lines are parallel but may be distinct

Question 17

If $$\vec{a} = \langle 1, -2, 3 \rangle$$ and $$\vec{b} = \langle 2, 1, 1 \rangle$$, what is the area of the triangle formed by these vectors?

Accepted Answer

$$\dfrac{5\sqrt{3}}{2}$$

Question 18

Which of the following is the correct way to check if two vectors $$\vec{v}$$ and $$\vec{w}$$ are orthogonal?

Accepted Answer

$$\vec{v} \cdot \vec{w} = 0$$

Vectors and Lines in Multivariable Calculus: Study Notes

Study Guide - Practice Questions

Study Guide - Flashcards