The dot product
For vectors u and v, the dot product u·v = u₁v₁ + u₂v₂ + u₃v₃. The result is a scalar.
The dot product relates to the angle: u·v = |u||v|cos(θ). When zero, the vectors are orthogonal.
u · v = u₁v₁ + u₂v₂ + u₃v₃
Dot product formula.
Dot Product Calculator
Calculate the dot product of two vectors in 2D or 3D, with angle and orthogonality detection.
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Dimensions
Vector 1
Vector 2
What is the dot product?
The dot product (scalar product) multiplies corresponding components of two vectors and sums the results. It reveals how much two vectors point in the same direction.
Orthogonal vectors
When the dot product equals zero the vectors are perpendicular (orthogonal), meaning they share no directional component.
Result
24
The dot product of (3, 4) and (4, 3) is 24. The angle between them is 16.2602°.
- Dot Product
- 24
- Angle (degrees)
- 16.2602°
- Angle (radians)
- 0.2838 rad
- Orthogonal
- No
- |v1| Magnitude
- 5
- |v2| Magnitude
- 5
Step-by-step
- Component products
- 3 × 4 = 12, 4 × 3 = 12
- Sum (dot product)
- 12 + 12 = 24
- |v1|
- √(3² + 4²) = 5
- |v2|
- √(4² + 3²) = 5
- cos θ
- 24 / (5 × 5) = 0.96
- θ
- 16.2602° (0.2838 rad)
Also in Linear Algebra
Linear Algebra
Dot product calculator: compute the dot product and angle between vectors
A dot product calculator multiplies corresponding components of two vectors and sums the results, also computing the angle between the vectors.
Frequently asked questions
The angle between the vectors is greater than 90° — they point in generally opposite directions.
The dot product returns a scalar measuring parallelism. The cross product returns a vector measuring perpendicularity.
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