Relative velocity in 2D

Intermediate

Vector subtraction of velocities.

Key relationv_AB = v_A − v_B

In two dimensions, v_AB = v_A − v_B is a vector subtraction. Components are subtracted separately, or a triangle of velocities is drawn.

Worked example

Car A 20 m/s east and car B 15 m/s north: relative velocity of A with respect to B is (20, −15), magnitude 25 m/s.

Common mistake

Remember to subtract the reference body's velocity, not add.

In more detail

Relative velocity in two dimensions is found by subtracting vectors: vAB equals vA minus vB, using components along each axis.

A boat crossing a river computes the boat's velocity relative to the bank.
Pilots correct for crosswinds by computing ground velocity from airspeed and wind.
Air traffic controllers track the velocity of one aircraft relative to another in two dimensions to keep a safe separation.

How do I subtract vectors?

Subtract their x components and their y components separately.

Does direction matter?

Yes, the result is a vector with both magnitude and direction.

How do I subtract two velocity vectors?

Find the components along each axis, subtract them, and then combine the results using Pythagoras and an angle.