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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