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 a pure orientation in space—it tells you where you are heading, not how far you will go.

Common stuck point: Opposite directions are genuinely different: north and south are not the same direction.

Sense of Study hint: Try pointing your arm in the direction described. Then ask: if I reversed my arm, would it still be the same direction?

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?

Solution

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

Answer

\theta = 45°
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?

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.
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Background Knowledge

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

orientation