Calculates the dot (scalar) product of two three-dimensional vectors.
How it works
Matching components from both vectors are multiplied together, then all three products are summed.
What this does not include
This does not include the cross product, which produces a new perpendicular vector rather than a single number — for that, use this site’s cross product calculator instead.
How to use this calculator
- Enter both vectors’ three components.
A worked example
Vectors (1, 2, 3) and (4, 5, 6): dot product = 1×4 + 2×5 + 3×6 = 32.
Vectors (1, 0, 0) and (0, 1, 0): dot product = 0 — the two vectors are perpendicular.
What the variables mean
| Variable | Meaning |
|---|---|
| Vector A (a1, a2, a3) | First 3D vector’s components |
| Vector B (b1, b2, b3) | Second 3D vector’s components |
Edge cases worth knowing
A dot product of zero means the vectors are perpendicular — the second example above is the textbook case, using the standard x and y unit vectors.
The dot product returns a single number, not a vector — unlike the cross product, which combines two vectors into a third vector rather than a scalar.
Frequently asked questions
What does a dot product of zero mean?
The two vectors are perpendicular to each other — a zero dot product is the standard test for perpendicularity.
Does the order of the vectors matter?
No — unlike the cross product, the dot product gives the same result regardless of which vector comes first.
What’s a real-world use for the dot product?
Calculating work done by a force, finding the angle between two vectors, and projecting one vector onto another all rely on the dot product.
