Direction Examples in Math

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)(1, 1). What angle does it make with the positive x-axis?

Answer

θ=45°\theta = 45°

First step

1
Step 1: The angle θ\theta satisfies tan⁡θ=y/x=1/1=1\tan\theta = y/x = 1/1 = 1.

Full solution

  1. 2
    Step 2: θ=arctan⁡(1)=45°\theta = \arctan(1) = 45°.
  2. 3
    Step 3: Since the point is in the first quadrant, the angle is 45°45° measured counterclockwise from the positive x-axis.
Direction is independent of magnitude — both (1,1)(1,1) and (100,100)(100, 100) point in the same direction (45°). The angle is found using θ=arctan⁡(y/x)\theta = \arctan(y/x), with the quadrant determining the correct branch.

Example 2

medium
Two vectors: a⃗=(1,0)\vec{a} = (1, 0) (east) and b⃗=(0,−1)\vec{b} = (0, -1) (south). What is the angle between them?

Example 3

medium
Convert the vector ⟨3,4⟩\langle 3, 4 \rangle into a magnitude and a direction angle from the positive xx-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)(2, 4) and (1,2)(1, 2) point in the same direction? Explain.

Example 2

hard
A ship sails N30°EN30°E (30° east of north). Express this direction as a unit vector in the standard (x,y)(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 ⟨5,0⟩\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∘45^\circ (northeast). Its components are equal. If its magnitude is 2\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 ⟨2,3⟩\langle 2, 3 \rangle and ⟨4,6⟩\langle 4, 6 \rangle point the same way?

Example 13

medium
Do ⟨1,2⟩\langle 1, 2 \rangle and ⟨−2,−4⟩\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 xx-axis, counterclockwise) of the vector ⟨0,−3⟩\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 ⟨3,4⟩\langle 3, 4 \rangle and ⟨8,6⟩\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 ⟨4,3⟩\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 ⟨0,5⟩\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∘360^\circ' a valid direction? If so, what is it?

Example 27

easy
What compass direction has bearing 180∘180^\circ?

Example 28

medium
Find a unit vector in the direction of ⟨6,8⟩\langle 6, 8 \rangle.

Example 29

medium
Are ⟨5,12⟩\langle 5, 12 \rangle and ⟨10,24⟩\langle 10, 24 \rangle parallel (same direction)?

Example 30

medium
Find the angle from the positive xx-axis (CCW) of ⟨−1,1⟩\langle -1, 1 \rangle.

Example 31

medium
Convert a bearing of N60∘EN60^\circ E to a standard angle from the positive xx-axis (east) measured counterclockwise.

Example 32

medium
Do vectors u⃗=⟨2,3⟩\vec{u} = \langle 2, 3 \rangle and v⃗=⟨3,2⟩\vec{v} = \langle 3, 2 \rangle point in the same direction?

Example 33

medium
A 100 N force acts 30∘30^\circ above the horizontal. Find its horizontal and vertical components.

Example 34

medium
A bearing of 315∘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)A = (1, 2) to B=(4,6)B = (4, 6).

Example 37

hard
Vector a⃗\vec{a} points at 45∘45^\circ above east with magnitude 2\sqrt{2}. Find its components.

Example 38

hard
The dot product u⃗⋅v⃗\vec{u}\cdot\vec{v} depends on the angle between the vectors. If u⃗⋅v⃗=0\vec{u}\cdot\vec{v} = 0, what does that say about their directions?

Example 39

hard
Decompose the velocity v⃗=⟨5,12⟩\vec{v} = \langle 5, 12 \rangle into magnitude and direction (angle from positive xx-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 u⃗=⟨1,0⟩\vec{u} = \langle 1, 0 \rangle and v⃗=⟨1,3⟩\vec{v} = \langle 1, \sqrt{3} \rangle.

Example 42

challenge
A pilot flies on a heading of N40∘EN40^\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 ⟨x,y,z⟩\langle x, y, z \rangle has only 2 degrees of freedom even though it has 3 components.
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Background Knowledge

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

orientation