Vector Dot Product Calculator

Dot Product Formula:

\[ \vec{a} \cdot \vec{b} = \sum_{i=1}^n a_i b_i = a_1b_1 + a_2b_2 + \cdots + a_nb_n \]

Dot Product Result:

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1. What is the Dot Product?

The dot product (or scalar product) is an algebraic operation that takes two equal-length sequences of numbers (vectors) and returns a single number. It's calculated by multiplying corresponding entries and summing those products.

2. How Does the Calculator Work?

The calculator uses the dot product formula:

\[ \vec{a} \cdot \vec{b} = \sum_{i=1}^n a_i b_i \]

Where:

\( \vec{a} \) and \( \vec{b} \) — Vectors of the same dimension
\( a_i \) and \( b_i \) — Components of the vectors
\( n \) — Dimension of the vectors

Explanation: The calculator takes two comma-separated vectors, splits them into arrays, multiplies corresponding components, and sums the results.

3. Applications of Dot Product

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.

4. Using the Calculator

Tips: Enter vectors as comma-separated values (e.g., "1,2,3"). Both vectors must have the same number of dimensions. The calculator will automatically handle the calculation.

5. Frequently Asked Questions (FAQ)

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 (in 3D) gives a vector perpendicular to both input vectors.

Q5: How is dot product used in machine learning?
A: It's used to calculate similarity between feature vectors, in neural networks for weighted sums, and in many algorithms like SVM and k-means.