Dot Product Calculator
Fast & Accurate Scalar Results for Vectors
Dot Product Calculator
Vector Dimension:
3
Vector A
Vector B
What is the Dot Product?
The dot product (also known as scalar product or inner product) is a mathematical operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation combines two vectors to produce a scalar quantity.
A · B = |A| × |B| × cos(θ)
Where |A| and |B| are the magnitudes of vectors A and B, and θ is the angle between them.
Algebraic Definition
For two vectors A = (a₁, a₂, ..., aₙ) and B = (b₁, b₂, ..., bₙ), the dot product is calculated as:
A · B = a₁b₁ + a₂b₂ + ... + aₙbₙ
Example Calculation
Let's calculate the dot product of A = (2, 3, 1) and B = (4, -2, 5):
A · B = (2 × 4) + (3 × -2) + (1 × 5) = 8 - 6 + 5 = 7
Properties of the Dot Product
- Commutative: A · B = B · A
- Distributive: A · (B + C) = A · B + A · C
- Scalar Multiplication: (cA) · B = c(A · B) = A · (cB)
- Orthogonality: If A · B = 0, then A and B are perpendicular (orthogonal)
Applications of Dot Product
The dot product has numerous applications in mathematics, physics, and engineering:
- Finding the angle between two vectors
- Projecting one vector onto another
- Determining if two vectors are perpendicular
- Calculating work in physics (work = force · displacement)
- Computer graphics and 3D rendering
- Machine learning and data analysis