Quick answer
The dot product is a · b = Σ aᵢbᵢ. For (1, 2, 3) · (4, 5, 6) it is 4 + 10 + 18 = 32.
Enter two vectors of the same dimension. The calculator returns the dot product.
Related tools: Cross Product Calculator: Cross product of two 3-D vectors. Slope Calculator: Slope, angle, distance and line equation through two points. LCM Calculator: Least common multiple of two or more numbers.
How to use the Dot Product Calculator
- Type vector A, for example 1, 2, 3.
- Type vector B, for example 1, 2, 3.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Vector A | Text, such as 1, 2, 3 | — |
| Vector B | Text, such as 1, 2, 3 | — |
Formula and method
Formula
a · b = Σ aᵢbᵢ
IEEE-754 double precision; shown to at most 6 decimal places.
Sum of component products.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Vector A = 1, 2, 3; Vector B = 4, 5, 6.
| Result | Value |
|---|---|
| Dot product | 32 |
Example 2
Inputs: Vector A = 1, 0; Vector B = 0, 1.
| Result | Value |
|---|---|
| Dot product | 0 |
The first line is the headline result (dot product). IEEE-754 double precision; shown to at most 6 decimal places.
Geometric meaning
a · b = |a||b| cos θ, so a zero dot product means the vectors are perpendicular. A positive value means an acute angle and a negative value an obtuse one.
Applications
Work in physics is force dotted with displacement, and projections in graphics use dot products. The result is a number, unlike the cross product, which is a vector.
What the sign tells you
For A = (1, 2, 3) and B = (4, 5, 6) the dot product is 1·4 + 2·5 + 3·6 = 32. A positive value means the vectors point in broadly the same direction, zero means they are perpendicular, and a negative value means they point apart.
The angle follows from cos θ = A·B ÷ (|A||B|). Here |A| ≈ 3.742 and |B| ≈ 8.775, so cos θ ≈ 0.9746 and θ ≈ 12.9°.
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Vector B: entering "1, 2, 3" shows "Both vectors need the same number of components." beside that field and no result.
Choosing between the Dot Product Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Cross Product Calculator](/math/cross-product-calculator/) | vector A and vector B | Cross product |
| [Slope Calculator](/math/slope-calculator/) | x₁, y₁ and x₂ | Slope |
| [LCM Calculator](/math/lcm-calculator/) | numbers | Least common multiple |
| [Factor Calculator](/math/factor-calculator/) | number | Factors |
| [Quadratic Formula Calculator](/math/quadratic-formula-calculator/) | a, b and c | Roots |
Background: algebra
Many algebra calculators rest on the same ideas: write the problem in a standard form, identify the coefficients, apply the formula and check the answer by substituting it back. For a quadratic ax² + bx + c = 0 the roots are (−b ± √(b² − 4ac)) ÷ 2a, and the discriminant b² − 4ac decides the kind of roots: positive gives two real roots, zero gives one repeated root and negative gives two complex conjugate roots.
If a = 0 the equation is linear, not quadratic. A sign slip on the −b term shifts both roots by the same amount, so substitute your roots back to check.
Assumptions and limitations
- Same dimension
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for vector B to confirm this.
Frequently asked questions
What is the dot product of perpendicular vectors?
The dot product of perpendicular vectors is zero, since cos 90° = 0.
Is the dot product commutative?
Yes. a · b = b · a, because multiplication of components is commutative.
How do I find the angle?
θ = arccos(a·b / |a||b|).
Do both vectors need the same number of components?
Yes. Each component of one vector is multiplied by the matching component of the other, so the dimensions must match.
How is the dot product different from the cross product?
The dot product returns a single number, while the cross product (in three dimensions) returns a vector perpendicular to both inputs.
What does the Dot Product Calculator calculate?
Dot product of two vectors. It returns Dot product.
What formula does the Dot Product Calculator use?
a · b = Σ aᵢbᵢ. Sum of component products.
What do I need to enter in the dot Product Calculator?
You enter Vector A and Vector B.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Same dimension.
What is an example of a dot Product result?
With vector A 1, 2, 3 and vector B 4, 5, 6, the dot product is 32.
Related calculators
- Cross Product Calculator — Cross product of two 3-D vectors.
- Slope Calculator — Slope, angle, distance and line equation through two points.
- LCM Calculator — Least common multiple of two or more numbers.
- Factor Calculator — List every factor of a positive whole number.
- Quadratic Formula Calculator — Solve ax² + bx + c = 0, including complex roots.
- Log Calculator — Logarithm of a number to any base.
- Solve for Unknown Fraction (Cross Multiplication) — Solve a/b = c/d when one value is unknown.
- Euclidean Algorithm Calculator — GCD with every division step shown.
- Common Multiple Calculator — First common multiples of two numbers.
Sources
- OpenStax, Precalculus 2e (Rice University): vectors, polar coordinates and trigonometry
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.