Question 1

Which of the following is the correct formula for the distance between two vectors $\vec{u} = (u_1, u_2)$ and $\vec{v} = (v_1, v_2)$ in $\mathbb{R}^2$?

Accepted Answer

$d(\vec{u}, \vec{v}) = \sqrt{(u_1 - v_1)^2 + (u_2 - v_2)^2}$

Question 2

Given the vector $\vec{a} = (3, -4)$, what is its magnitude (length)?

Accepted Answer

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

Question 3

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

Accepted Answer

The zero vector has all components equal to zero and its magnitude is $0$.

Question 4

If $\vec{u} = (2, 1)$ and $\vec{v} = (4, 2)$, which of the following is TRUE?

Accepted Answer

$\vec{u}$ and $\vec{v}$ are collinear because $\vec{v} = 2\vec{u}$.

Question 5

Which of the following vectors is a unit vector?

Accepted Answer

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

Question 6

Let $\vec{a} = (2, -3)$ and $\vec{b} = (4, -6)$. Which statement is correct?

Accepted Answer

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

Question 7

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

Accepted Answer

$x^2 + y^2 = 1$

Question 8

Given $\vec{u} = (1, 2, 3)$ and $\vec{v} = (-1, 0, 1)$, what is the distance between $\vec{u}$ and $\vec{v}$?

Accepted Answer

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

Question 9

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

Accepted Answer

A scalar is a real number that can multiply a vector, changing its magnitude but not its direction (unless the scalar is negative).

Question 10

If $\vec{u} = (a, b)$ and $\vec{v} = (ka, kb)$ for some real number $k 
eq 0$, what is the relationship between $\vec{u}$ and $\vec{v}$?

Accepted Answer

$\vec{u}$ and $\vec{v}$ are collinear.

Question 11

Which of the following is the correct formula for the length (magnitude) of a vector $\vec{v} = (v_1, v_2, v_3)$ in $\mathbb{R}^3$?

Accepted Answer

$\|\vec{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2}$

Question 12

Suppose $\vec{u} = (2, 3)$ and $\vec{v} = (4, 6)$. What is the unit vector in the same direction as $\vec{v}$?

Accepted Answer

$\left( \frac{2}{\sqrt{13}}, \frac{3}{\sqrt{13}} \right)$

Question 13

If $\vec{a} = (1, 2)$ and $\vec{b} = (2, 4)$, and $\vec{c} = (3, 6)$, which statement is TRUE?

Accepted Answer

All three vectors are collinear.

Question 14

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

Accepted Answer

It is a unit vector.

Question 15

If $\vec{u} = (x, y)$ and $\vec{v} = (y, x)$, for which values of $x$ and $y$ are $\vec{u}$ and $\vec{v}$ collinear (other than the zero vector)?

Accepted Answer

For $x = y 
eq 0$ or $x = -y 
eq 0$

Question 16

Given $\vec{u} = (1, 2, 3)$, what is the length of $-2\vec{u}$?

Accepted Answer

$2\sqrt{14}$

Question 17

Which of the following best describes when two vectors in $\mathbb{R}^n$ are collinear?

Accepted Answer

They are scalar multiples of each other.

Question 18

If $\vec{u} = (a, b)$ is a unit vector and $a > 0$, $b > 0$, which of the following could be $\vec{u}$?

Accepted Answer

$\left( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right)$

Question 19

Suppose $\vec{u} = (2, -1)$ and $\vec{v} = (k, -0.5)$ are collinear. What is the value of $k$?

Accepted Answer

$k = 4$

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

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