Dot Product Formula:
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The dot product (also called scalar product) is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It measures the similarity between two vectors.
The calculator uses the dot product formula:
Where:
Explanation: The dot product is calculated by multiplying corresponding components of the vectors and then summing those products.
Details: The dot product is used in physics to calculate work done, in computer graphics for lighting calculations, in machine learning for similarity measures, and in many other mathematical and engineering applications.
Tips: Enter vectors as comma-separated values (e.g., "1,2,3" for a 3D vector). Both vectors must have the same number of elements. The calculator will automatically handle the calculation.
Q1: What's the geometric interpretation of dot product?
A: The dot product equals the product of the vectors' magnitudes and the cosine of the angle between them: \( \vec{A} \cdot \vec{B} = \|\vec{A}\| \|\vec{B}\| \cos(\theta) \).
Q2: What does a dot product of zero mean?
A: A zero dot product indicates the vectors are perpendicular (orthogonal) to each other.
Q3: Can I calculate dot product for vectors of different dimensions?
A: No, dot product is only defined for vectors of the same dimension.
Q4: What's the difference between dot product and cross product?
A: Dot product gives a scalar result, while cross product (only in 3D) gives a vector perpendicular to both input vectors.
Q5: How is dot product used in machine learning?
A: It's used in calculating similarity between feature vectors, in neural networks for weighted sums, and in many other algorithms.