Math

Instant, private, and free

Dot Product Calculator.

Calculate the dot product of two 2D or 3D vectors. Leave z components at zero for 2D.

On-device calculationNo signup
01

Set your values

Results update as you type.

dotProduct: 5

dotProduct

0.000000

Results update automatically as you type.

Use Cases

Check if vectors are perpendicular

If the dot product is zero, the vectors are orthogonal (90° apart). Useful in geometry and physics.

Example: A = (1, 0), B = (0, 1) → dot product = 0, so they are perpendicular.

Compute work done by a force

In physics, work is the dot product of force and displacement vectors. This calculator helps you find that scalar quickly.

Example: Force F = (5, 3) N, displacement d = (2, 1) m → work = 5*2 + 3*1 = 13 J.

Frequently Asked Questions

What is the dot product of two vectors?
The dot product (also called scalar product) is a single number obtained by multiplying corresponding components of two vectors and summing the results. For 2D vectors (a1, a2) and (b1, b2), it's a1*b1 + a2*b2. For 3D, add a3*b3.
How do I use this calculator for 2D vectors?
Enter the x and y components for both vectors A and B. Leave the z fields empty or set to 0. The calculator will compute the dot product using the 2D formula.
What does the dot product tell me?
The dot product indicates the relationship between two vectors: positive if they point in similar directions, negative if opposite, and zero if they are perpendicular. It's also used to find the angle between vectors and in physics for work calculations.

Tips & Common Mistakes

Tips

  • For 2D vectors, leave the z fields blank or set them to 0 to avoid confusion.
  • Double-check the sign of each component; a negative component affects the result.
  • Use the dot product to find the angle between vectors: cos θ = dot product / (|A| * |B|).
  • If the dot product is zero, the vectors are perpendicular (orthogonal).

Common Mistakes to Avoid

  • Forgetting to include the z component when using 3D vectors, leading to an incorrect result.
  • Mixing up the order of vectors; the dot product is commutative, so order doesn't matter, but ensure you pair corresponding components.
  • Using the dot product to determine if vectors are parallel; a zero dot product means perpendicular, not parallel.

Last updated: August 13, 2026