Q: Given points $$P = (1, 2)$$ and $$Q = (3, -1) in$$ $$\mathbb{R}^2$$, what is the direction vector $$\vec{d} = \overrightarrow{PQ}$$?

$$\begin{bmatrix} 2 \\ -3 \end{bmatrix}$$

Q: Which of the following is the correct definition of the transpose of a vector $$\vec{x} in$$ $$\mathbb{R}^n$$?

If $$\vec{x} = \begin{bmatrix} x_1$ \\ x_2$ \\ \vdots \\ x_n \end{bmatrix}$$, then $$\vec{x}^T = [x_1$, x_2$, \ldots, x_n]$$

Q: Which of the following is a standard (elementary) vector in $$\mathbb{R}^3$$?

$$\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$$

Q: If $$\vec{x} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$$ and $$\vec{y} = \begin{bmatrix} -1 \\ 3 \end{bmatrix}$$, what is $$\vec{x} + \vec{y}$$?

$$\begin{bmatrix} 1 \\ 6 \end{bmatrix}$$

Q: Let $$\vec{x} = \begin{bmatrix} 1 \\ -2 \\ 3 \end{bmatrix}$$ and $$t = -2. $$What is $$t\vec{x}$$?

$$\begin{bmatrix} -2 \\ 4 \\ -6 \end{bmatrix}$$

Q: If $$\vec{x} = \begin{bmatrix} 3 \\ -1 \end{bmatrix}$$ and $$\vec{y} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$$, what is $$\vec{x} - \vec{y}$$?

$$\begin{bmatrix} 2 \\ -3 \end{bmatrix}$$

Q: Which of the following is the correct set notation for $$\mathbb{R}^n$$?

$$\left\{ \begin{bmatrix} x_1$ \\ x_2$ \\ \vdots \\ x_n \end{bmatrix} : x_1$, x_2$, \ldots, x_n \in \mathbb{R} \right\}$$

Q: If $$\vec{x} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$$ and $$\vec{y} = \begin{bmatrix} 3 \\ -1 \end{bmatrix}$$, which of the following is the result of $$2\vec{x} - \vec{y}$$?

$$\begin{bmatrix} -1 \\ 5 \end{bmatrix}$$

Question 1

Which of the following best describes a vector in $$\mathbb{R}^3$$?

Accepted Answer

A vector with three real number components, represented as $$\begin{bmatrix} x_1$ \ x_2$ \ x_3$ \end{bmatrix}$$

Question 2

Given points $$P = (1, 2)$$ and $$Q = (3, -1) in$$ $$\mathbb{R}^2$$, what is the direction vector $$\vec{d} = \overrightarrow{PQ}$$?

Accepted Answer

$$\begin{bmatrix} 2 \ -3 \end{bmatrix}$$

Question 3

Which of the following is the correct definition of the transpose of a vector $$\vec{x} in$$ $$\mathbb{R}^n$$?

Accepted Answer

If $$\vec{x} = \begin{bmatrix} x_1$ \ x_2$ \ \vdots \ x_n \end{bmatrix}$$, then $$\vec{x}^T = [x_1$, x_2$, \ldots, x_n]$$

Question 4

Which of the following is a standard (elementary) vector in $$\mathbb{R}^3$$?

Accepted Answer

$$\begin{bmatrix} 0 \ 1 \ 0 \end{bmatrix}$$

Question 5

If $$\vec{x} = \begin{bmatrix} 2 \ 3 \end{bmatrix}$$ and $$\vec{y} = \begin{bmatrix} -1 \ 3 \end{bmatrix}$$, what is $$\vec{x} + \vec{y}$$?

Accepted Answer

$$\begin{bmatrix} 1 \ 6 \end{bmatrix}$$

Question 6

Which property of vector addition is illustrated by the equation $$\vec{x} + \vec{y} = \vec{y} + \vec{x}$$?

Accepted Answer

Commutative property

Question 7

Let $$\vec{x} = \begin{bmatrix} 1 \ -2 \ 3 \end{bmatrix}$$ and $$t = -2. $$What is $$t\vec{x}$$?

Accepted Answer

$$\begin{bmatrix} -2 \ 4 \ -6 \end{bmatrix}$$

Question 8

Suppose $$\vec{x} = \begin{bmatrix} 2 \ 4 \ 5 \end{bmatrix} in$$ $$\mathbb{R}^3. $$Which point in space does this vector represent if its initial point is at the origin?

Accepted Answer

$$(2, 4, 5)$$

Question 9

If $$\vec{x} = \begin{bmatrix} 3 \ -1 \end{bmatrix}$$ and $$\vec{y} = \begin{bmatrix} 1 \ 2 \end{bmatrix}$$, what is $$\vec{x} - \vec{y}$$?

Accepted Answer

$$\begin{bmatrix} 2 \ -3 \end{bmatrix}$$

Question 10

Which of the following statements about the zero vector $$\vec{0} in$$ $$\mathbb{R}^n is $$true?

Accepted Answer

$$\vec{x} + \vec{0} = \vec{x}$$ for any $$\vec{x} in$$ $$\mathbb{R}^n$$

Question 11

Which of the following is the correct set notation for $$\mathbb{R}^n$$?

Accepted Answer

$$\left\{ \begin{bmatrix} x_1$ \ x_2$ \ \vdots \ x_n \end{bmatrix} : x_1$, x_2$, \ldots, x_n \in \mathbb{R} ight\}$$

Question 12

If $$\vec{x} = \begin{bmatrix} 1 \ 2 \end{bmatrix}$$ and $$\vec{y} = \begin{bmatrix} 3 \ -1 \end{bmatrix}$$, which of the following is the result of $$2\vec{x} - \vec{y}$$?

Accepted Answer

$$\begin{bmatrix} -1 \ 5 \end{bmatrix}$$

Question 13

Which of the following is NOT a property of vector addition in $$\mathbb{R}^n$$?

Accepted Answer

Multiplicative inverse property

Question 14

If $$\vec{x} = \begin{bmatrix} 2 \ -5 \end{bmatrix}$$ and $$\vec{y} = \begin{bmatrix} 3 \ 1 \end{bmatrix}$$, what is $$\vec{x} - \vec{y}$$?

Accepted Answer

$$\begin{bmatrix} -1 \ -6 \end{bmatrix}$$

Question 15

Which of the following vectors in $$\mathbb{R}^3 is $$parallel to $$\vec{a} = \begin{bmatrix} 2 \ -1 \ 4 \end{bmatrix}$$?

Accepted Answer

$$\begin{bmatrix} 4 \ -2 \ 8 \end{bmatrix}$$

Question 16

A real-world application: In physics, the displacement from point $$A = (0, 0, 0) to $$point $$B = (5, -2, 3) is $$represented by which vector?

Accepted Answer

$$\begin{bmatrix} 5 \ -2 \ 3 \end{bmatrix}$$

Question 17

Which of the following is the correct result of $$-2\vec{x} if$$ $$\vec{x} = \begin{bmatrix} 4 \ -3 \end{bmatrix}$$?

Accepted Answer

$$\begin{bmatrix} -8 \ 6 \end{bmatrix}$$

Question 18

If $$\vec{x} = \begin{bmatrix} 1 \ 0 \ 0 \end{bmatrix}$$, $$\vec{y} = \begin{bmatrix} 0 \ 1 \ 0 \end{bmatrix}$$, and $$\vec{z} = \begin{bmatrix} 0 \ 0 \ 1 \end{bmatrix} in$$ $$\mathbb{R}^3$$, what is $$\vec{x} + \vec{y} + \vec{z}$$?

Accepted Answer

$$\begin{bmatrix} 1 \ 1 \ 1 \end{bmatrix}$$

Vectors and Euclidean Space: Foundations and Operations

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