Direction Math Example 1

Follow the full solution, then compare it with the other examples linked below.

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?

Solution

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

Answer

θ=45°\theta = 45°
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.

About Direction

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

Learn more about Direction →

More Direction Examples

Example 2 medium

Two vectors: [formula] (east) and [formula] (south). What is the angle between them?

Example 3 easy

Do the vectors [formula] and [formula] point in the same direction? Explain.

Example 4 hard

A ship sails [formula] (30° east of north). Express this direction as a unit vector in the standard

← All Direction Examples Direction Concept