Dot Product Calculator
Solve dot product problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Dot Product Calculator
Calculator
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Formula: A·B = a₁b₁ + a₂b₂ + a₃b₃ | Angle θ = arccos(A·B / (|A| × |B|))
Worked example — A·B = 32, angle ≈ 12.93°
Formula
A·B = a₁b₁ + a₂b₂ + a₃b₃ | Angle θ = arccos(A·B / (|A| × |B|))
The dot product (scalar product) of two vectors equals the sum of products of corresponding components. It also equals |A||B|cos(θ), where θ is the angle between them. Dot product = 0 means vectors are perpendicular.
Worked Examples
Example 1: 3D vectors
Problem:A = (1,2,3), B = (4,5,6)
Solution:Dot = 1×4 + 2×5 + 3×6 = 4 + 10 + 18 = 32. Angle ≈ 12.93°
Result:A·B = 32, angle ≈ 12.93°
Frequently Asked Questions
What does the dot product represent?
Geometrically, A·B = |A||B|cos(θ). It measures how much one vector projects onto another. Positive = same general direction, negative = opposite, zero = perpendicular.
Dot product vs cross product?
Dot product gives a scalar (number). Cross product (3D only) gives a vector perpendicular to both inputs with magnitude |A||B|sin(θ). Dot product measures alignment; cross product measures perpendicularity.
How is the dot product used in machine learning and AI?
The dot product is one of the most fundamental operations in machine learning. Neural network layers compute weighted sums using dot products between input vectors and weight matrices. In natural language processing, word embeddings and sentence embeddings are compared using cosine similarity, which normalizes the dot product by the product of the vector magnitudes. Attention mechanisms in transformers (the foundation of large language models) compute dot products between query and key vectors to determine which parts of the input to focus on. High dot product values indicate high similarity or relevance between vectors.
What is the vector projection formula using the dot product?
The scalar projection of vector A onto vector B is the dot product A·B divided by the magnitude of B, giving the length of the shadow A casts along B. The vector projection multiplies this scalar by the unit vector of B, giving the actual projected vector component along B's direction. Projections are used in physics to decompose forces into components along an inclined plane, in computer graphics for lighting and shading calculations, and in statistics for finding the portion of variation explained by a particular direction in principal component analysis.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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