Dot Product Calculator
Compute the dot (inner) product of two vectors with geometric interpretation.
Inputs
A · B
32.000000
Angle Between (degrees)
12.9332
Orthogonal?
No
Step by step
A
= [1, 2, 3]
B
= [4, 5, 6]
A · B = Σ AᵢBᵢ
= 32
||A||
= 3.741657
||B||
= 8.774964
θ = arccos(A·B / (||A||·||B||))
= 12.9332°
How it works
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
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.