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 $$P = (-2, 1, 4) to$$ $$Q = (2, -5, 1)$$?

Accepted Answer

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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