Question 1

Given a vector $\vec{v} = \langle 3, 4 \rangle$, what is the magnitude of $\vec{v}$?

Accepted Answer

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

Question 2

Which formula correctly calculates the magnitude of a vector $\vec{v} = \langle a, b \rangle$?

Accepted Answer

$|\vec{v}| = \sqrt{a^2 + b^2}$

Question 3

A displacement vector in a city grid is $\vec{d} = \langle -6, 8 \rangle$. What is the magnitude of this displacement?

Accepted Answer

$\sqrt{(-6)^2 + 8^2} = 10$

Question 4

If $\vec{u} = \langle 1, 2 \rangle$ and $\vec{v} = \langle 4, -2 \rangle$, what is the magnitude of $\vec{u} + \vec{v}$?

Accepted Answer

$\sqrt{(1+4)^2 + (2+(-2))^2} = \sqrt{25} = 5$

Question 5

A vector $\vec{a}$ has components $\langle x, y \rangle$. If $|\vec{a}| = 13$ and $x = 5$, what is $y$?

Accepted Answer

$y = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$

Question 6

Which of the following best describes the geometric meaning of the magnitude of a vector $\vec{v} = \langle a, b \rangle$?

Accepted Answer

It is the length of the vector from the origin to the point $(a, b)$.

Question 7

A vector $\vec{w}$ has magnitude $7$ and points in the direction $\langle 3, 4 \rangle$. What is the unit vector in the direction of $\vec{w}$?

Accepted Answer

$\left\langle \frac{3}{5}, \frac{4}{5} \right\rangle$

Question 8

A boat travels east for 6 km and then north for 8 km. What is the straight-line distance from its starting point?

Accepted Answer

$\sqrt{6^2 + 8^2} = 10$ km

Question 9

If $\vec{p} = \langle x, y \rangle$ and $|\vec{p}| = \sqrt{50}$, which of the following could be the values of $x$ and $y$?

Accepted Answer

$x = 5, y = 5$

Question 10

A vector $\vec{q}$ has components $\langle -7, 24 \rangle$. What is $|\vec{q}|$?

Accepted Answer

$\sqrt{(-7)^2 + 24^2} = 25$

Question 11

A drone flies 9 meters east and 12 meters north. What is the magnitude of its displacement vector?

Accepted Answer

$\sqrt{9^2 + 12^2} = 15$ meters

Question 12

If $\vec{r} = \langle 0, y \rangle$ and $|\vec{r}| = 8$, what is $y$?

Accepted Answer

$y = 8$

Question 13

Which of the following vectors has a magnitude of $\sqrt{2}$?

Accepted Answer

$\langle 1, 1 \rangle$

Question 14

A vector $\vec{s}$ has components $\langle 8, 15 \rangle$. What is $|\vec{s}|$?

Accepted Answer

$\sqrt{8^2 + 15^2} = 17$

Question 15

If $\vec{t} = \langle a, b \rangle$ and $|\vec{t}| = 10$, which equation must be true?

Accepted Answer

$a^2 + b^2 = 100$

Question 16

A hiker walks 5 km west and 12 km south. What is the magnitude of the hiker's displacement?

Accepted Answer

$\sqrt{5^2 + 12^2} = 13$ km

Question 17

Which of the following is NOT a correct way to find the magnitude of $\vec{v} = \langle a, b \rangle$?

Accepted Answer

$|\vec{v}| = a + b$

Vectors in the Coordinate Plane: Magnitude and Applications

Study Guide - Practice Questions