Resultant Velocity Calculator (add 2 or 3 velocity vectors: speed and direction)

Understanding the resultant velocity calculator — concept and principle

This calculator adds two or three velocities as vectors and gives the resultant speed and direction, its east and north components, its angle from the first velocity and, for comparison, the plain sum of speeds. It is used for boats crossing rivers, aircraft in wind and physics problems on relative motion.

Perpendicular speeds add like the sides of a right triangle: 3 and 4 make 5, not 7.

How it is calculated

v_x = Σ v cos α; v_y = Σ v sin α

Resultant speed = √(v_x² + v_y²)

Direction = atan2(v_y, v_x), in degrees anticlockwise from east

Speeds can be in any unit, as long as all are the same. Directions use the maths convention, not compass bearings.

The principle

Velocities have both size and direction, so they add head-to-tail like arrows, not like plain numbers. Splitting each velocity into east and north components turns the problem into two ordinary sums. The length of the resulting arrow is the speed, by Pythagoras, and its angle is the direction. This is how relative motion works: a boat's velocity relative to the shore is its velocity through the water plus the water's velocity.

Worked example

Default inputs: 4 at 90° (north) and 3 at 0° (east).

x-component = 4 cos 90° + 3 cos 0° = 3; y-component = 4 sin 90° + 3 sin 0° = 4.
Resultant speed = √(3² + 4²) = 5.
Direction = atan2(4, 3) = 53.1° from east, which is 36.9° east of north.
Angle from velocity 1 = 53.1° − 90° = −36.9°; plain sum of speeds = 7.

Each point is the calculator re-run with only “Velocity 1 – direction (° from east, anticlockwise)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Velocity 1 – direction (° from east, anticlockwise)	Resultant speed
22.5	6.868
45	6.478
67.5	5.847
90	5
112	3.977
135	2.834
180	1

How to read the result

The resultant is never more than the sum of speeds and never less than their difference. Here a boat heading north at 4 m/s in a current flowing east at 3 m/s actually moves at 5 m/s, drifting downstream. To travel straight across, the boat must point partly upstream.

Common mistakes

Adding speeds directly.
Mixing compass bearings with mathematical angles.
Using degrees in a calculator set to radians.

Keep in mind: The result assumes constant velocities. Real currents and winds vary in space and time.

Frequently asked questions

How do I convert a compass bearing to this angle?

Angle = 90° − bearing, then add 360° if negative.

Can the resultant be zero?

Yes, if the velocities cancel exactly.

Why is the resultant 5 and not 7?

The two velocities are at right angles, so they combine by Pythagoras.

How do I cross a river straight?

Point upstream so that the current component is cancelled.

Sources: University of Central Florida PHY2048, chapter 4 lecture notes (relative velocity)

Checked 9 Nov 2026. Relative-velocity addition v_PA = v_PB + v_BA (University of Central Florida PHY2048 chapter 4 lecture notes, Eq. 15.4) and vector components (standard method).

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