Vector Calculator (Dot Product)
Compute the dot product, magnitudes and the angle between 2D or 3D vectors with fully substituted working. Includes the cosθ = (a·b) ÷ (|a||b|) step and perpendicular detection via a·b = 0 — handy for math and physics.
a = (a₁, a₂, a₃)
Components of vector a
b = (b₁, b₂, b₃)
Components of vector b
Try an example
Enter the components of a and b to see the dot product, magnitudes and angle.
Done
Vector Calculator (Dot Product)
How to use
- 1
Pick the dimension
Choose 2D (two components) or 3D (three).
- 2
Enter the components
Fill in the components of vectors a and b. Perpendicular and parallel examples are included.
- 3
Read values and working
The dot product, magnitudes and angle appear with substituted formulas. a·b = 0 means perpendicular.
Features
- Dot products of 2D and 3D vectors with substituted component-form working
- Magnitudes |a| and |b| plus the angle from cosθ in degrees and radians
- Sign-based classification: acute, perpendicular (a·b = 0) or obtuse
- Clear handling of zero vectors; inputs are never transmitted
Use cases
Check math homework
Compare hand-worked dot products and angles with the substituted steps.
Physics work calculations
Find the angle between force and displacement for W = F·s·cosθ.
Test perpendicularity
Confirm a·b = 0 for perpendicular vectors quickly and numerically.
Details
The dot product works component-wise, a·b = a₁b₁ + a₂b₂ (+ a₃b₃), and the magnitude is |a| = √(a₁² + a₂² + a₃²). The angle θ between vectors follows from rearranging the definition a·b = |a||b|cosθ into cosθ = (a·b) ÷ (|a||b|). The tool substitutes your components into every one of these formulas.
The sign of the dot product reveals how two vectors relate: a·b > 0 means an acute angle (pointing in similar directions), a·b = 0 means perpendicular, and a·b < 0 means obtuse. In particular a·b = 0 is the standard perpendicularity test, and it appears in physics as a force doing zero work when it acts perpendicular to the displacement.
Conditions: the angle is defined only when both |a| and |b| are greater than zero — the angle against a zero vector is undefined and reported as such. Components accept plain numbers with absolute values up to one million in 2D or 3D. Inputs are never transmitted; all computation happens locally in your browser.
FAQ
What is a dot product?
It is an operation that turns two vectors into the single number |a||b|cosθ. Component-wise it is a·b = a₁b₁ + a₂b₂ (+ a₃b₃), and both formulas give the same value. The result is a number, not a vector.
How do I find the angle between vectors?
Rearrange the definition a·b = |a||b|cosθ into cosθ = (a·b) ÷ (|a||b|) and apply the inverse cosine. The tool shows the substitution through to the angle in degrees and radians.
What does a·b = 0 mean?
It means the angle between the vectors is 90° — they are perpendicular. If either vector has magnitude zero the angle is undefined, so such a pair is not called perpendicular.
What does the sign of the dot product tell me?
a·b > 0 gives an acute angle (0°–90°), a·b = 0 is perpendicular, and a·b < 0 is obtuse (90°–180°) — one number that summarizes how much two directions agree.
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Verified: Known values ((3,4)·(4,3)=24, 3D dot 20, perpendicular 90°, parallel 0°), zero-vector handling and the English page are covered by browser tests
Did you know?
A zero dot product means perpendicular — the same test that games and architecture use to check how light strikes a surface, and why a force perpendicular to motion does no work.