Symbolab Dot Product Calculator

Dot Product Formula:

\[ \vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3 \]

Vector u (x component):

Vector u (y component):

Vector u (z component):

Vector v (x component):

Vector v (y component):

Vector v (z component):

Dot Product (u · v):

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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 (usually coordinate vectors) and returns a single number. It measures the magnitude of one vector in the direction of another.

2. How Does the Calculator Work?

The calculator uses the dot product formula:

\[ \vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3 \]

Where:

\( \vec{u} \) — First vector with components (u₁, u₂, u₃)
\( \vec{v} \) — Second vector with components (v₁, v₂, v₃)
The result is a scalar quantity

Explanation: The dot product is calculated by multiplying corresponding components and summing 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 engineering for projection calculations.

4. Using the Calculator

Tips: Enter the components of both vectors. For 2D vectors, leave the z-components blank. The calculator handles both 2D and 3D vectors.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between dot product and cross product?
A: Dot product returns a scalar quantity, while cross product returns a vector perpendicular to both input vectors.

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 dot product be negative?
A: Yes, a negative dot product indicates the angle between vectors is greater than 90 degrees.

Q4: 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.

Q5: How is dot product used in machine learning?
A: It's used to calculate similarity between feature vectors, such as in cosine similarity measures.