Dot Product Calculator — Vector Dot Product & Angle

The Formula

The dot product (also called the scalar product or inner product) is an algebraic operation that takes two equal-length sequences of numbers and returns a single number. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. The dot product is zero when the vectors are orthogonal (perpendicular), positive when they point in similar directions (angle < 90°), and negative when they point in opposite directions (angle > 90°).

Variable Definitions

How to Use This Calculator

1. 1

   Enter the components of the first vector (a) and second vector (b). The z-components are optional (default 0).

2. 2

   The calculator computes the dot product, magnitudes, and the angle between the vectors.

3. 3

   Check the "Orthogonal?" result — if "Yes", the dot product is zero and the vectors are perpendicular.

4. 4

   Review the scalar projection, which gives the length of the shadow of vector a cast onto vector b.

Quick Reference

Common Applications

• Physics: computing work done by a force (W = F · d), where only the force component in the direction of motion contributes.
• Computer graphics: implementing the Phong lighting model where the dot product of the surface normal and light direction determines brightness.
• Machine learning: calculating cosine similarity between document vectors for information retrieval and recommendation systems.
• Game development: detecting whether a character is facing toward or away from an object using the dot product of direction vectors.
• Linear algebra: testing orthogonality, computing projections, and implementing the Gram-Schmidt process for creating orthonormal bases.

The dot product measures how much two vectors align with each other, reaching maximum when parallel and zero when perpendicular

Understanding the Concept

The dot product is the simplest and most fundamental vector multiplication operation. Unlike scalar multiplication which scales a vector, the dot product combines two vectors to produce a scalar. The algebraic definition (sum of component-wise products) is easy to compute, while the geometric interpretation (|a||b|cos θ) reveals its true meaning: it measures how much one vector extends in the direction of another. When the dot product is zero, the vectors are orthogonal — a property used constantly in graphics, physics, and linear algebra. A positive dot product means the vectors are generally pointing in the same direction; a negative one means they point oppositely. The scalar projection (a · b / |b|) gives the signed length of the orthogonal projection of a onto the line through b. This is used extensively in computing components of forces, decomposing vectors into basis directions, and in the Gram-Schmidt orthogonalization process.

Frequently Asked Questions

Sources & References