Question 1

Which of the following best describes a vector in $\mathbb{R}^2$?

Accepted Answer

A directed line segment with both magnitude and direction, represented as an ordered pair $(a, b)$ where $a$ and $b$ are real numbers.

Question 2

Given vectors $\vec{u} = (2, 3)$ and $\vec{v} = (4, 6)$, which statement is true?

Accepted Answer

$\vec{u}$ and $\vec{v}$ are collinear because $\vec{v}$ is a scalar multiple of $\vec{u}$.

Question 3

What is the length (magnitude) of the vector $\vec{a} = (3, -4)$?

Accepted Answer

$\sqrt{3^2 + (-4)^2} = 5$

Question 4

Which of the following is a unit vector?

Accepted Answer

$(\frac{3}{5}, \frac{4}{5})$

Question 5

If $\vec{u} = (1, 2)$ and $\vec{v} = (4, 8)$, what is the value of $k$ such that $\vec{v} = k\vec{u}$?

Accepted Answer

$k = 4$

Question 6

What is the distance between the vectors $\vec{a} = (1, 2)$ and $\vec{b} = (4, 6)$ in $\mathbb{R}^2$?

Accepted Answer

$\sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = 5$

Question 7

Which of the following statements about the zero vector $\vec{0}$ in $\mathbb{R}^2$ is correct?

Accepted Answer

The zero vector has no direction and its magnitude is zero.

Question 8

If $\vec{a} = (2, -3)$, what is $-\frac{1}{2}\vec{a}$?

Accepted Answer

$(-1, 1.5)$

Question 9

Which of the following is the correct formula for the distance between two vectors $\vec{a} = (a_1, a_2)$ and $\vec{b} = (b_1, b_2)$ in $\mathbb{R}^2$?

Accepted Answer

$d(\vec{a}, \vec{b}) = \sqrt{(a_1 - b_1)^2 + (a_2 - b_2)^2}$

Question 10

Suppose $\vec{u} = (x, y)$ is a unit vector. Which equation must be satisfied?

Accepted Answer

$x^2 + y^2 = 1$

Question 11

If $\vec{a} = (3, 4)$, what is the unit vector in the same direction as $\vec{a}$?

Accepted Answer

$(\frac{3}{5}, \frac{4}{5})$

Question 12

Which of the following statements about scalar multiplication of vectors is correct?

Accepted Answer

Multiplying a vector by a negative scalar reverses its direction.

Question 13

Given $\vec{a} = (2, 1)$ and $\vec{b} = (4, 2)$, which of the following is true?

Accepted Answer

$\vec{b}$ is a scalar multiple of $\vec{a}$, so they are collinear.

Question 14

If $\vec{u} = (x, y)$ and $\vec{v} = (y, x)$, for which values of $x$ and $y$ are $\vec{u}$ and $\vec{v}$ equal?

Accepted Answer

When $x = y$

Question 15

A vector $\vec{a}$ is said to be collinear with $\vec{b}$ if:

Accepted Answer

There exists a scalar $k$ such that $\vec{a} = k\vec{b}$

Question 16

If $\vec{a} = (1, 2, 3)$ and $\vec{b} = (-1, 0, 1)$, what is the distance between $\vec{a}$ and $\vec{b}$ in $\mathbb{R}^3$?

Accepted Answer

$\sqrt{(1 - (-1))^2 + (2 - 0)^2 + (3 - 1)^2} = \sqrt{4 + 4 + 4} = 2\sqrt{3}$

Question 17

Which of the following is NOT a property of the zero vector $\vec{0}$?

Accepted Answer

It is a unit vector.

Question 18

If $\vec{a} = (2, -1)$, what is the magnitude of $-3\vec{a}$?

Accepted Answer

$3\sqrt{5}$

Question 19

Which of the following best describes a scalar in the context of vectors?

Accepted Answer

A real number that can multiply a vector, changing its magnitude and possibly its direction.

Question 20

Given $\vec{a} = (2, 3)$ and $\vec{b} = (4, 6)$, what is the unit vector in the direction of $\vec{a} + \vec{b}$?

Accepted Answer

$(\frac{6}{\sqrt{45}}, \frac{9}{\sqrt{45}})$

Vectors: Definitions, Properties, and Applications (Math 1229A/B Unit 1)

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