Q: Which of the following vectors is a unit vector?

$\begin{bmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \end{bmatrix}$

Q: Given $\vec{a} = \begin{bmatrix} 2 \\ -1 \\ 2 \end{bmatrix}$ and $\vec{b} = \begin{bmatrix} 1 \\ 3 \\ 0 \end{bmatrix}$, what is the dot product $\vec{a} \cdot \vec{b}$?

$2 \times 1 + (-1) \times 3 + 2 \times 0 = -1$

Q: Given $\vec{x} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$, what is the unit vector in the direction of $\vec{x}$?

$\begin{bmatrix} 3/5 \\ 4/5 \end{bmatrix}$

Question 1

Let $\vec{x} = \begin{bmatrix} 6 \ 8 \end{bmatrix}$. What is the norm (length) of $\vec{x}$?

Accepted Answer

$\sqrt{36 + 64} = 10$

Question 2

Which of the following vectors is a unit vector?

Accepted Answer

$\begin{bmatrix} 1/\sqrt{2} \ 1/\sqrt{2} \end{bmatrix}$

Question 3

Given $\vec{a} = \begin{bmatrix} 2 \ -1 \ 2 \end{bmatrix}$ and $\vec{b} = \begin{bmatrix} 1 \ 3 \ 0 \end{bmatrix}$, what is the dot product $\vec{a} \cdot \vec{b}$?

Accepted Answer

$2 	imes 1 + (-1) 	imes 3 + 2 	imes 0 = -1$

Question 4

What is the length of the vector from point $P = (-2, 1, 4)$ to $Q = (2, -5, 1)$?

Question 5

Given $\vec{x} = \begin{bmatrix} 3 \ 4 \end{bmatrix}$, what is the unit vector in the direction of $\vec{x}$?

Accepted Answer

$\begin{bmatrix} 3/5 \ 4/5 \end{bmatrix}$

Question 6

If $A = \begin{bmatrix} 2 & 5 \ 1 & 3 \end{bmatrix}$, what is $\det(A)$?

Accepted Answer

$(2)(3) - (5)(1) = 6 - 5 = 1$

Question 7

Let $\vec{a} = \begin{bmatrix} 1 \ 2 \end{bmatrix}$ and $\vec{b} = \begin{bmatrix} 3 \ 4 \end{bmatrix}$. What is the Gram determinant $G(\vec{a}, \vec{b})$?

Accepted Answer

$\|\vec{a}\|^2 \|\vec{b}\|^2 - (\vec{a} \cdot \vec{b})^2 = (1^2 + 2^2)(3^2 + 4^2) - (1 	imes 3 + 2 	imes 4)^2 = (5)(25) - (11)^2 = 125 - 121 = 4$

Question 8

Which of the following statements about the Cauchy-Schwarz inequality is correct?

Accepted Answer

$|\vec{a} \cdot \vec{b}| \leq \|\vec{a}\| \|\vec{b}\|$ for all $\vec{a}, \vec{b} \in \mathbb{R}^n$

Question 9

Given $\vec{x} = \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix}$, which formula gives the norm of $\vec{x}$?

Accepted Answer

$\sqrt{x_1^2 + x_2^2 + x_3^2}$

Question 10

If $\vec{a} = \begin{bmatrix} 0 \ 1 \end{bmatrix}$ and $\vec{b} = \begin{bmatrix} 1 \ 0 \end{bmatrix}$, what is the angle between $\vec{a}$ and $\vec{b}$?

Accepted Answer

$\frac{\pi}{2}$ radians

Question 11

A vector $\vec{v}$ in $\mathbb{R}^n$ is called a unit vector if:

Accepted Answer

$\|\vec{v}\| = 1$

Question 12

Given points $A = (1, 2)$ and $B = (5, 6)$, what is the midpoint $M$ of segment $AB$?

Accepted Answer

$\left( \frac{1+5}{2}, \frac{2+6}{2} \right) = (3, 4)$

Question 13

If $\vec{x} = \begin{bmatrix} 2 \ 1 \ 4 \ 2 \end{bmatrix}$, what is $\|\vec{x}\|^2$?

Accepted Answer

$2^2 + 1^2 + 4^2 + 2^2 = 4 + 1 + 16 + 4 = 25$

Question 14

Which of the following is the correct formula for the determinant of a $2 	imes 2$ matrix $A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}$?

Accepted Answer

$a_{11}a_{22} - a_{12}a_{21}$

Question 15

Suppose $\vec{a}$ and $\vec{b}$ are nonzero vectors in $\mathbb{R}^n$. When is the Gram determinant $G(\vec{a}, \vec{b})$ equal to zero?

Accepted Answer

When $\vec{a}$ and $\vec{b}$ are linearly dependent

Question 16

If $\vec{a} = \begin{bmatrix} 1 \ 2 \end{bmatrix}$ and $\vec{b} = \begin{bmatrix} 2 \ 4 \end{bmatrix}$, what is the angle between $\vec{a}$ and $\vec{b}$?

Accepted Answer

$0$ radians

Question 17

Given $\vec{x} = \begin{bmatrix} x_1 \ x_2 \ \vdots \ x_n \end{bmatrix}$, which of the following is always true?

Accepted Answer

$\|\vec{x}\| \geq 0$

Norm, Unit Vectors, and Gram Determinant in College Algebra

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