Dot Product Calculator

What is the Dot Product?

The dot product (also called the scalar product or inner product) is a fundamental operation between two vectors that produces a scalar value. Unlike the cross product which gives a vector result, the dot product returns a single number that represents the geometric relationship between two vectors.

Dot Product Formula

For two vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), the dot product is:

a · b = a₁b₁ + a₂b₂ + a₃b₃

Alternatively, using the geometric definition:

a · b = |a| × |b| × cos(θ)

Where θ is the angle between the two vectors.

Finding the Angle Between Vectors

The dot product is commonly used to find the angle between two vectors:

cos(θ) = (a · b) / (|a| × |b|)

θ = arccos((a · b) / (|a| × |b|))

How to Use This Calculator

1. Enter Vector A Components: Input the x, y, and z values for the first vector.
2. Enter Vector B Components: Input the x, y, and z values for the second vector.
3. View Results: The calculator instantly shows the dot product, magnitudes, angle between vectors, and projections.

Properties of the Dot Product

• Commutative: a · b = b · a
• Distributive: a · (b + c) = a · b + a · c
• Scalar Multiplication: (ka) · b = k(a · b)
• Orthogonal Vectors: If a · b = 0, the vectors are perpendicular (90°)
• Parallel Vectors: If a · b = |a||b|, the vectors point in the same direction (0°)
• Anti-parallel Vectors: If a · b = -|a||b|, the vectors point in opposite directions (180°)

Applications of the Dot Product

• Physics: Calculating work done (W = F · d)
• Computer Graphics: Lighting calculations and shading
• Machine Learning: Similarity measures and cosine similarity
• Projection: Finding the component of one vector along another
• Angle Detection: Determining if vectors are parallel, perpendicular, or at an angle

Scalar and Vector Projection

The scalar projection of a onto b is:

projₛ(a onto b) = (a · b) / |b|

The vector projection of a onto b is:

projᵥ(a onto b) = ((a · b) / |b|²) × b

Frequently Asked Questions

What does a zero dot product mean?

A dot product of zero indicates the two vectors are perpendicular (orthogonal) to each other, meaning they meet at a 90° angle.

What's the difference between dot product and cross product?

The dot product produces a scalar (single number) and measures how much vectors point in the same direction. The cross product produces a vector perpendicular to both inputs and measures the area of the parallelogram they form.

Can I use the dot product with 2D vectors?

Yes! Simply use a · b = a₁b₁ + a₂b₂. This calculator supports both 2D (leave z as 0) and 3D vectors.

How do I know if vectors point in the same direction?

If the dot product is positive, they generally point in the same direction (angle < 90°). If negative, they point in opposite directions (angle > 90°).

Understanding Your Results

Dot Product (Scalar Result)

The primary result is a single number. A positive value means vectors point in the same general direction, while a negative value means they point away from each other.

Angle Between Vectors

This tells you exactly how much the vectors diverge. 0° means they are identical in direction, 90° means they are perfectly perpendicular, and 180° means they point in opposite directions.

Projections

The scalar projection tells you the "length" of Vector A that lies along Vector B. The vector projection provides this as an actual vector in 3D space.

Important Limitations

• This calculator assumes standard Euclidean space.
• Floating point precision may result in very small non-zero values for theoretically zero results.
• Vectors must have the same number of components (3D in this tool).

Disclaimer: This calculator is for educational and informational purposes. While highly accurate for mathematical calculations, it should not be the sole basis for critical engineering or scientific decisions without verification.