2D vector calculator — dot product, angle, projection and more

How to use the 2D vector calculator

1. Enter vector u as u x and u y.
2. Enter vector v as v x and v y.
3. Read the Dot product (u·v) first. Below it are the Cross product (2D scalar, u×v), the lengths |u| and |v|, u + v and u − v.
4. Read Angle between, Unit vector of u and Projection of u on v.

What it does and when to use it

Vectors describe anything with a size and a direction: a force, a velocity, a displacement on a map, or a direction in a game. The dot product and angle tell you how much two directions agree, which matters for work done by a force or lighting in graphics. The projection splits one vector into the part along another, such as the part of gravity acting down a slope. The cross-product sign tells a program whether to turn left or right.

For vectors with a z component, use the 3D vector calculator. The length |u| is the same calculation as the distance between two points from the origin. The trigonometry calculator helps when you start from a length and an angle.

How it works

For u = (u.x, u.y) and v = (v.x, v.y):

• Dot product: u·v = u.x × v.x + u.y × v.y
• 2D cross product: u×v = u.x × v.y − u.y × v.x
• Magnitude: |u| = √(u.x² + u.y²)
• Sum and difference: add or subtract matching components
• Angle: θ = arccos(u·v ÷ (|u| × |v|))
• Unit vector: u ÷ |u|
• Projection of u on v: (u·v ÷ |v|²) × v

Worked examples

Angle between two vectors, step by step. u = (2, 3) and v = (5, −1).

1. Dot product: 2 × 5 + 3 × (−1) = 10 − 3 = 7.
2. Lengths: |u| = √(4 + 9) = √13 = 3.605551275 and |v| = √(25 + 1) = √26 = 5.099019514.
3. cos θ = 7 ÷ (3.605551275 × 5.099019514) = 7 ÷ 18.38477631 = 0.3807498053.
4. θ = arccos(0.3807498053).

The tool shows Angle between: 67.61986495° and Cross product (2D scalar, u×v): -17. The negative sign means v lies clockwise from u.

Splitting a vector. u = (6, 8) and v = (1, 0), a unit vector along the x-axis. The tool gives Projection of u on v: (6, 0), the horizontal part of u. The unit vector of u is (0.6, 0.8), and the angle is 53.13010235°.

Limits and tips

• Only two components per vector. Points are not vectors: subtract start from end first if you have coordinates.
• Results show up to 10 significant figures. Keep extra digits if you reuse a result in a later step.
• There are no units; components can be in any unit, as long as both vectors use the same one.
• Everything runs in your browser.