Dot Product Calculator: Angle and Projection
This dot product calculator multiplies two vectors of any length, from 2D and 3D up to n dimensions. Enter the components separated by commas, such as 1, 2, 3 and 4, 5, 6. You get the dot product a · b with every multiplication shown, the magnitude of each vector, the angle between them in degrees and radians, the scalar and vector projection of a onto b, and whether the vectors are orthogonal (perpendicular), parallel or neither. It is useful for physics work problems, computer graphics and linear algebra homework.
Dot Product Calculator
Find the dot product of two vectors in any dimension, plus the angle, projection and orthogonality check.
How to use the Dot Product Calculator
- Enter vector a, with components separated by commas.
- Enter vector b with the same number of components.
- Read the dot product and the steps.
- Check the angle, projection and orthogonality.
Dot product formula
a · b = a₁b₁ + a₂b₂ + … + aโbโ = |a||b| cos ฮธ
Features
- Any dimension (2D, 3D, n-D)
- Steps shown
- Angle in degrees and radians
- Scalar and vector projection
- Orthogonal and parallel check
Frequently asked questions
How do you calculate a dot product?
Multiply matching components and add the results: (1, 2, 3) · (4, 5, 6) = 1·4 + 2·5 + 3·6 = 32.
How do I find the angle between two vectors?
Use cos ฮธ = (a · b) ÷ (|a| |b|). For (1, 2, 3) and (4, 5, 6): 32 ÷ (3.742 × 8.775) = 0.9746, so ฮธ ≈ 12.93°.
What does a dot product of zero mean?
The vectors are orthogonal (perpendicular), as long as neither is the zero vector.
What is the scalar projection?
The length of a's shadow on b: (a · b) ÷ |b|. The vector projection is ((a · b) ÷ |b|²) × b.
Can a dot product be negative?
Yes. A negative dot product means the angle between the vectors is more than 90°.