Dot Product Calculator
Compute the dot (scalar) product of two equal-length vectors and see each component-wise contribution.
Sum of component-wise products
- 1
Component a₁ × b₁
1 × 4 = 4 - 2
Component a₂ × b₂
2 × -5 = -10 - 3
Component a₃ × b₃
3 × 6 = 18 - 4
Dot product (sum of products)
4 + -10 + 18 = 12
How does this calculator work?
The dot product of two equal-length vectors is the sum of their component-wise products: a · b = Σ aᵢbᵢ. Enter both vectors using spaces or commas, and the result is a single scalar. A zero result means the vectors are perpendicular; positive means an acute angle, negative an obtuse one.
Formula
How this is calculated
Enter two vectors, a and b, as lists of numbers separated by spaces or commas. Each vector is parsed into its components a = (a₁, …, aₙ) and b = (b₁, …, bₙ). The two vectors must contain the same number of components; otherwise the dot product is undefined and an error is shown.
The dot product is computed by multiplying corresponding components and summing the results: a · b = Σ aᵢbᵢ. The result is a single scalar number. The breakdown shows each individual product aᵢbᵢ, so you can see exactly how positive and negative terms combine to form the total.
The dot product is dimensionless in the sense that it returns a plain scalar regardless of how many dimensions the vectors have. Geometrically, a · b = |a||b|cos θ, so a result of zero means the vectors are orthogonal (perpendicular), a positive result means the angle between them is acute, and a negative result means it is obtuse.
Frequently asked questions
The dot product is only defined for vectors with the same number of components. If the lengths differ, the calculator shows a warning instead of a result.
A dot product of zero means the two vectors are orthogonal (perpendicular) to each other, since a · b = |a||b|cos θ and cos 90° = 0.
Yes. Components can be any real numbers, including negatives and decimals. Just separate each value with a space or comma.
Also known as
TG we-Calculate Editorial Team. (2026). Dot Product Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/dot-product-calculator
TG we-Calculate Editorial Team. "Dot Product Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/dot-product-calculator.
TG we-Calculate Editorial Team, "Dot Product Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/dot-product-calculator
@misc{wecalculate_dot_product_calculator, title = {Dot Product Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/dot-product-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?