Dot Product Calculator
Compute the dot (inner) product of two vectors with geometric interpretation.
About Dot Product Calculator
The dot product (inner product) of two vectors A and B is the sum of the products of corresponding components: A·B = Σ AᵢBᵢ. Geometrically, A·B = ||A|| ||B|| cos θ where θ is the angle between them. Two vectors are orthogonal (perpendicular) if and only if their dot product is zero.
Formulas
Algebraic Form
A · B = Σᵢ AᵢBᵢ
- A
- First vector
- B
- Second vector
Geometric Form
A · B = ||A|| · ||B|| · cos(θ)
- θ
- Angle between A and B
Worked example
With the default values below, Dot Product Calculator returns A · B of 32.000000. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Vector A (comma or space separated)
- 1, 2, 3
- Vector B (comma or space separated)
- 4, 5, 6
Results
- A · B
- 32.000000
- Angle Between (degrees)
- 12.9332
- Orthogonal?
- No
Step by step
- 1A= [1, 2, 3]
- 2B= [4, 5, 6]
- 3A · B = Σ AᵢBᵢ= 32
- 4||A||= 3.741657
- 5||B||= 8.774964
- 6θ = arccos(A·B / (||A||·||B||))= 12.9332°
Assumptions and limits
- The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
When is the dot product zero?
When the vectors are perpendicular (orthogonal). This is the standard test for orthogonality in any dimension.
Can the dot product be negative?
Yes. It is negative when the angle between the vectors exceeds 90°, meaning they point in generally opposite directions.