How to use it
- Choose the Dimension: 2D or 3D.
- Choose the Operation. Then type the components of vector a in Vector a (components separated by spaces), and of vector b or c where the page asks for it.
- Read Result. For the dot and cross product, Check says whether the vectors are perpendicular or parallel.
- Go through How it is worked out to see each component.
Dot, cross and triple products
The dot product of two vectors is the sum of the products of their components, a · b = a₁b₁ + a₂b₂ + a₃b₃. It is zero when the vectors are perpendicular. The cross product of two 3D vectors is a vector perpendicular to both, with components (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). The triple product a · (b × c) is the determinant of the matrix with rows a, b and c.
a · b = a₁b₁ + a₂b₂ + a₃b₃
a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)- The length of a is |a| = √(a · a), and the unit vector a ÷ |a| has length 1.
- The angle between a and b follows from cos θ = (a · b) ÷ (|a| |b|).
- The projection of a onto b is ((a · b) ÷ (b · b)) b, and its scalar component is (a · b) ÷ |b|.
Exactness and what the calculator does not cover
Components are read exactly, so the dot, cross and triple products and the projection are exact. A length is kept as √ of a whole or fraction number where that is possible, and the angle is a decimal because the arccosine of most values is not a simple fraction of π.
- Vectors in 2D and 3D only; the cross product in 2D is the single number a₁b₂ − a₂b₁.
- Complex components, vectors in four or more dimensions, and matrix operations are not covered; use the matrix calculator for matrices.
- A zero vector has no direction, so the unit vector, angle and projection onto it are refused.
Worked example: the cross product of (1, 2, 3) and (4, −5, 6)
The first component of a × b is 2 × 6 − 3 × (−5) = 12 + 15 = 27, the second is 3 × 4 − 1 × 6 = 6 and the third is 1 × (−5) − 2 × 4 = −13. So a × b = (27, 6, −13). As a check, a · (27, 6, −13) = 27 + 12 − 39 = 0, so the cross product is perpendicular to a.
The dot product a · b is 4 − 10 + 18 = 12, and the angle between the vectors follows from cos θ = 12 ÷ (√14 × √77), about 0.3655, which is 68.56°.
| Result | |
|---|---|
| Cross product a × b | (27, 6, −13) |
| Dot product a · b | 12 |
| Length of a | √14 ≈ 3.7417 |
| Angle between a and b | 68.56° (1.1966 rad) |
Frequently asked questions
What does a dot product of zero tell me?
It means the two vectors are perpendicular, so they meet at a right angle. The page writes this as Perpendicular in the check box, whenever the dot product comes out exactly 0.
Why is the cross product not a number in 3D?
The cross product of two 3D vectors is another vector, perpendicular to both. In 2D the same idea gives a single number, a₁b₂ − a₂b₁, and the page shows that number.
How do I type a vector?
Type its components in order, separated by spaces: 1 2 3 for the vector with x = 1, y = 2 and z = 3. Negative components need a minus sign, and decimals use your language's decimal mark.
Why does a length such as √14 appear instead of 3.74?
The square root is kept exact, so the result is √14 and its decimal approximation is shown after it. Only the decimal is rounded, never the calculation itself.
Are the vectors I type sent to a server?
No. The calculation runs in your browser, and nothing you type is sent to us.
Sources
- Wikipedia: Dot product
- Wikipedia: Cross product
- Wikipedia: Triple product
- Wikipedia: Vector projection
Figures and text last checked on Tuesday, November 10, 2026.