Direction Examples in Math: 47 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Direction.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The orientation of movement or facing in space, independent of speed or distance—where something points.

North, south, east, west—or the way an arrow points, regardless of how long the arrow is.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Direction is pure orientation — the way something points, with speed and distance stripped out.

Common stuck point: The procedure for direction is the easy part; the trap is mixing distance into direction. Asking "Am I describing only which way something points, with distance and speed set aside?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I describing only which way something points, with distance and speed set aside?

Worked Examples

Example 1

easy

A vector points from the origin to the point

(1, 1)

. What angle does it make with the positive x-axis?

Answer

\theta = 45°

First step

Step 1: The angle

\theta

satisfies

\tan\theta = y/x = 1/1 = 1

Full solution

2
Step 2: $\theta = \arctan(1) = 45°$ .
3
Step 3: Since the point is in the first quadrant, the angle is $45°$ measured counterclockwise from the positive x-axis.

Direction is independent of magnitude — both

(1,1)

and

(100, 100)

point in the same direction (45°). The angle is found using

\theta = \arctan(y/x)

, with the quadrant determining the correct branch.

Example 2

medium

Two vectors:

\vec{a} = (1, 0)

(east) and

\vec{b} = (0, -1)

(south). What is the angle between them?

Example 3

medium

Convert the vector

\langle 3, 4 \rangle

into a magnitude and a direction angle from the positive

x

-axis.

Example 4

medium

A boat heads east at 5 m/s; the current pushes it north at 3 m/s. Find the direction of the boat's resulting velocity.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Do the vectors

(2, 4)

and

(1, 2)

point in the same direction? Explain.

Example 2

hard

A ship sails

N30°E

(30° east of north). Express this direction as a unit vector in the standard

(x, y)

coordinate system where x is east and y is north.

Example 3

easy

Two cars both travel north, one at 30 mph and one at 60 mph. Do they have the same direction?

Example 4

easy

Which of these describes a direction: '10 km' or 'northeast'?

Example 5

easy

An arrow points due east. Another arrow, twice as long, also points due east. Same direction?

Example 6

easy

What is the opposite direction of 'north'?

Example 7

easy

Turning from north to east is a turn of how many degrees?

Example 8

easy

Is 'velocity' (speed with direction) more like direction alone or a combination of size and direction?

Example 9

easy

A vector

\langle 5, 0 \rangle

points in which direction?

Example 10

easy

Do two objects moving the same distance always move in the same direction?

Example 11

medium

A vector points at a bearing of

45^\circ

(northeast). Its components are equal. If its magnitude is

\sqrt{2}

, find the components.

Example 12

medium

Two vectors point in the same direction if one is a positive scalar multiple of the other. Do

\langle 2, 3 \rangle

and

\langle 4, 6 \rangle

point the same way?

Example 13

medium

\langle 1, 2 \rangle

and

\langle -2, -4 \rangle

point in the same or opposite directions?

Example 14

medium

A bearing is measured clockwise from north. What bearing is due west?

Example 15

medium

A unit vector represents pure direction. Why is its magnitude always 1?

Example 16

medium

A ship sails north, then turns to head east. Through what angle did its direction change?

Example 17

medium

Why does a round trip (out and back) result in zero net displacement even though you clearly traveled?

Example 18

medium

Find the angle (from the positive

x

-axis, counterclockwise) of the vector

\langle 0, -3 \rangle

Example 19

challenge

A plane's heading is northeast, but a strong wind from the north pushes it so its actual track is due east. Qualitatively, how must the pilot adjust the heading to keep traveling northeast?

Example 20

challenge

Two vectors

\langle 3, 4 \rangle

and

\langle 8, 6 \rangle

— do they point in the same direction? Justify with ratios.

Example 21

challenge

Why does specifying a direction in 3D space require two angles, while in 2D it needs only one?

Example 22

challenge

A robot moves

\langle 4, 3 \rangle

then wants to return straight home. In what direction (as a unit vector) must it travel?

Example 23

easy

A vector

\langle 0, 5 \rangle

points in which compass direction?

Example 24

easy

Two arrows have the same direction but different lengths. Are they the same vector?

Example 25

easy

Turning from east to north is a counterclockwise rotation of how many degrees?

Example 26

easy

Is a 'bearing of

360^\circ

' a valid direction? If so, what is it?

Example 27

easy

What compass direction has bearing

180^\circ

Example 28

medium

Find a unit vector in the direction of

\langle 6, 8 \rangle

Example 29

medium

Are

\langle 5, 12 \rangle

and

\langle 10, 24 \rangle

parallel (same direction)?

Example 30

medium

Find the angle from the positive

x

-axis (CCW) of

\langle -1, 1 \rangle

Example 31

medium

Convert a bearing of

N60^\circ E

to a standard angle from the positive

x

-axis (east) measured counterclockwise.

Example 32

medium

Do vectors

\vec{u} = \langle 2, 3 \rangle

and

\vec{v} = \langle 3, 2 \rangle

point in the same direction?

Example 33

medium

A 100 N force acts

30^\circ

above the horizontal. Find its horizontal and vertical components.

Example 34

medium

A bearing of

315^\circ

corresponds to which cardinal/intercardinal direction?

Example 35

hard

Two vectors point in different directions and have the same magnitude. Their sum's direction is what relative to each?

Example 36

hard

Find the unit vector pointing from

A = (1, 2)

B = (4, 6)

Example 37

hard

Vector

\vec{a}

points at

45^\circ

above east with magnitude

\sqrt{2}

. Find its components.

Example 38

hard

The dot product

\vec{u}\cdot\vec{v}

depends on the angle between the vectors. If

\vec{u}\cdot\vec{v} = 0

, what does that say about their directions?

Example 39

hard

Decompose the velocity

\vec{v} = \langle 5, 12 \rangle

into magnitude and direction (angle from positive

x

-axis).

Example 40

hard

Why does a unit vector in 2D have just one degree of freedom while a vector in 2D has two?

Example 41

hard

Find the angle between

\vec{u} = \langle 1, 0 \rangle

and

\vec{v} = \langle 1, \sqrt{3} \rangle

Example 42

challenge

A pilot flies on a heading of

N40^\circ E

at 200 km/h with a wind blowing from the west at 30 km/h. Express the ground velocity components (east, north).

Example 43

challenge

In 3D, a direction can be parameterized by two angles (e.g., azimuth and elevation). Explain why a unit vector

\langle x, y, z \rangle

has only 2 degrees of freedom even though it has 3 components.

← Back to Direction Practice Mode

Related Concepts

Orientation Vector Intuition Angles

Background Knowledge

These ideas may be useful before you work through the harder examples.

orientation