Dot Product Calculator
Calculate the dot product of two 2D or 3D vectors, plus magnitudes and the angle between them. Free online vector calculator.
How to use this dot product calculator
The dot product (scalar product) measures how aligned two vectors are. Positive means an acute angle, zero means perpendicular, and negative means obtuse. Leave z components at 0 for 2D work in the plane.
This vector calculator is built for physics (work = F · d), computer graphics (lighting, projection), and linear algebra homework. It reports A · B, both magnitudes, the angle in degrees, and whether the vectors are orthogonal.
Formula
Worked example
(1, 2, 3) · (4, 5, 6) = 4 + 10 + 18 = 32. |A| = √14, |B| = √77, so θ = arccos(32 / √(14×77)) ≈ 12.93°.
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Frequently asked questions
How do you calculate a dot product?
Multiply matching components and add: AₓBₓ + AᵧBᵧ + A_zB_z. In 2D omit the z terms or set them to 0.
How do I find the angle between two vectors?
θ = arccos((A · B) / (|A| |B|)). This calculator returns that angle in degrees.
What does a negative dot product mean?
The angle is greater than 90°. The vectors point into opposite half-spaces.
Can I use this for 2D vectors?
Yes. Set A_z and B_z to 0.
When are two vectors perpendicular?
When A · B = 0 and neither vector is zero.
Results are estimates for informational purposes and are not financial, legal, or professional advice. Last reviewed 2026-08-20.