Frequently Asked Questions
Find answers to common questions about dot product calculations, properties, and applications. For more detailed information, visit our definition page or try our interactive calculator.
What is the dot product of two vectors?
The dot product (also known as the scalar product) is an algebraic operation that takes two equal-length sequences of numbers (vectors) and returns a single number. For two vectors A and B, the dot product is calculated as A·B = AxBx + AyBy + AzBz in 3D space. Learn more on our definition page.
How is dot product different from cross product?
The dot product results in a scalar quantity, while the cross product results in a vector. The dot product measures the similarity of direction between two vectors, while the cross product produces a vector perpendicular to both input vectors. The dot product is commutative (A·B = B·A), while the cross product is anti-commutative (A×B = -B×A).
What does a dot product of zero mean?
A dot product of zero indicates that the vectors are orthogonal (perpendicular) to each other. This is a fundamental property used in geometry and physics to test for perpendicularity. See examples of orthogonal vectors in our examples section.
Can dot product be negative?
Yes, the dot product can be negative. This occurs when the angle between the vectors is greater than 90 degrees. A negative dot product indicates that the vectors are pointing in generally opposite directions. The range of dot product values depends on the vector magnitudes and the angle between them.
How is dot product used in real life?
Dot products have numerous real-world applications, including:
- Physics: Calculating work done by a force
- Computer Graphics: Lighting calculations and shading
- Engineering: Projecting forces onto specific directions
- Machine Learning: Measuring similarity between data points
- Economics: Calculating weighted averages
Learn more about these applications on our applications page.
What is the maximum value of a dot product?
The maximum possible value of a dot product occurs when two vectors point in exactly the same direction. In this case, the dot product equals the product of the magnitudes of the two vectors: A·B = ||A|| ||B||. The minimum value occurs when they point in opposite directions: A·B = -||A|| ||B||.
Still have questions? Try our interactive calculator to experiment with different vectors or explore more worked examples.
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