Vector Dot Product Calculator

Vector A

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Vector B

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Understanding Vector Dot Product

What is a Vector Dot Product?

The dot product (scalar product) of two vectors is defined as the product of their magnitudes and the cosine of the angle between them:

A · B = |A| |B| cos(θ)

Or in component form:

A · B = AxBx + AyBy + AzBz

where:

A and B are vectors
|A| and |B| are vector magnitudes
θ is the angle between vectors
Ax, Ay, Az are components of vector A
Bx, By, Bz are components of vector B

Properties of Dot Product

Key Properties:

Commutative: A · B = B · A
Distributive: A · (B + C) = A · B + A · C
Scalar Multiplication: (kA) · B = k(A · B)
Self Dot Product: A · A = |A|²

Special Cases:

Parallel vectors (θ = 0°): A · B = |A| |B|
Perpendicular vectors (θ = 90°): A · B = 0
Opposite vectors (θ = 180°): A · B = -|A| |B|

Geometric Applications

Finding Angles Between Vectors

θ = arccos((A · B)/(|A| |B|))

Vector Projection

proj_B A = ((A · B)/|B|²)B

Work in Physics

W = F · d = |F| |d| cos(θ)

Applications in Computer Graphics

Surface Normal Calculations
Light Intensity: I = L · N
Shadow Calculations