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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