cross product: meaning, definition, pronunciation and examples
C1/C2Formal/Technical
Quick answer
What does “cross product” mean?
A mathematical operation that takes two vectors and produces a third vector perpendicular to both, with magnitude equal to the product of their magnitudes and the sine of the angle between them.
Audio
Pronunciation
Definition
Meaning and Definition
A mathematical operation that takes two vectors and produces a third vector perpendicular to both, with magnitude equal to the product of their magnitudes and the sine of the angle between them.
In broader contexts, can refer to any product or outcome resulting from the combination or interaction of two distinct elements, systems, or disciplines.
Dialectal Variation
British vs American Usage
Differences
No significant lexical differences; both use 'cross product'. Some British texts historically used 'vector product' more frequently, but 'cross product' is now standard in both varieties.
Connotations
Identical technical connotations. Both associate it with mathematics, physics, and engineering contexts.
Frequency
Equally frequent in academic and technical writing in both varieties.
Grammar
How to Use “cross product” in a Sentence
the cross product of A and BA cross Bcompute the cross product between X and YVocabulary
Collocations
Examples
Examples of “cross product” in a Sentence
verb
British English
- To find the torque, you need to cross product the force vector with the radius vector.
- The software can cross product any two 3D arrays.
American English
- Cross product those two vectors to get the normal.
- The function will cross product the input matrices.
adverb
British English
- [Not used adverbially]
American English
- [Not used adverbially]
adjective
British English
- The cross-product result gave us the perpendicular vector.
- We examined the cross-product properties in detail.
American English
- The cross-product vector is orthogonal to both inputs.
- Cross-product calculations are fundamental to physics.
Usage
Meaning in Context
Business
Rare except in technical businesses (engineering firms, software development) where it might describe interdisciplinary projects.
Academic
Standard in mathematics, physics, engineering, and computer science textbooks and papers.
Everyday
Virtually never used in casual conversation.
Technical
Core term in vector mathematics, physics (torque, angular momentum), computer graphics, and robotics.
Vocabulary
Synonyms of “cross product”
Strong
Neutral
Weak
Vocabulary
Antonyms of “cross product”
Watch out
Common Mistakes When Using “cross product”
- Using 'cross product' for the dot/scalar product.
- Applying to vectors in 2D space (standard cross product requires 3D).
- Confusing with Cartesian product in set theory.
FAQ
Frequently Asked Questions
The cross product of two vectors results in another vector, while the dot product results in a scalar. The cross product also requires three-dimensional space.
The standard cross product as defined in vector algebra is only for three-dimensional vectors. However, in 2D, one often uses the scalar equivalent (the perpendicular dot product or '2D cross product') which gives a scalar representing the signed area.
The right-hand rule is a mnemonic to determine the direction of the resulting vector. If you point your index finger in the direction of the first vector and your middle finger in the direction of the second, your thumb will point in the direction of their cross product.
It is extensively used in physics (torque, Lorentz force), engineering (moments, electromagnetism), computer graphics (calculating surface normals, lighting), and robotics (angular velocity calculations).
A mathematical operation that takes two vectors and produces a third vector perpendicular to both, with magnitude equal to the product of their magnitudes and the sine of the angle between them.
Cross product is usually formal/technical in register.
Cross product: in British English it is pronounced /ˌkrɒs ˈprɒdʌkt/, and in American English it is pronounced /ˌkrɔːs ˈprɑːdʌkt/. Tap the audio buttons above to hear it.
Phrases
Idioms & Phrases
- “[No common idioms; technical term]”
Learning
Memory Aids
Mnemonic
Think of crossing your fingers: two fingers (vectors) pointing in different directions create a third direction (the cross product) perpendicular to both.
Conceptual Metaphor
MATHEMATICAL OPERATION IS PRODUCTION (producing a new vector); INTERDISCIPLINARY WORK IS CROSS-MULTIPLICATION (combining fields).
Practice
Quiz
Which of the following is a property of the cross product?