vector product
C1+Formal, Technical, Academic
Definition
Meaning
A binary operation on two vectors in three-dimensional space, resulting in a vector that is perpendicular to the plane containing the original vectors, with magnitude equal to the product of their magnitudes and the sine of the angle between them.
Also used in higher-dimensional contexts (e.g., 7D via octonions) and abstract algebra. In physics and engineering, it describes torques, angular momentum, and electromagnetic force.
Linguistics
Semantic Notes
Its result is a pseudovector (axial vector), meaning its direction depends on the handedness of the coordinate system (right-hand rule). It is anticommutative: a × b = -b × a.
Dialectal Variation
British vs American Usage
Differences
No significant lexical differences. "Vector product" is standard in both, though "cross product" is more common in US technical contexts.
Connotations
Identical technical connotations.
Frequency
"Cross product" is more frequent in American engineering and physics textbooks. "Vector product" is slightly more formal and common in pure mathematics texts in both varieties.
Vocabulary
Collocations
Grammar
Valency Patterns
The vector product of [NP] and [NP] is [NP/ADJ]To find [NP], take the vector product [PP][NP] is given by the vector product.Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Phrases
Idioms & Phrases
- “The right-hand rule for the vector product”
Usage
Context Usage
Business
Not used.
Academic
Core concept in linear algebra, vector calculus, physics, and engineering courses.
Everyday
Virtually never used.
Technical
Essential in mechanics (torque), electromagnetism (Lorentz force), computer graphics (surface normals), and robotics.
Examples
By Part of Speech
verb
British English
- To vector-product the two matrices, you need a three-dimensional representation.
- We need to vector-product the force and displacement vectors.
American English
- To cross-product the two vectors, apply the determinant formula.
- The software is designed to vector-multiply the inputs.
adverb
British English
- The forces were combined vector-product-wise.
- Multiply these vectorially, not scalar-wise.
American English
- The components were multiplied cross-product-style.
- Combine them vector-product-fashion to get the torque.
adjective
British English
- The vector-product calculation yielded an unexpected axial result.
- We examined the vector-product properties in detail.
American English
- The cross-product operation is fundamental here.
- He derived a new vector-product identity.
Examples
By CEFR Level
- In physics, the vector product helps find turning forces.
- The result of a vector product is another vector.
- The torque is defined as the vector product of the radius vector and the applied force.
- Unlike the dot product, the vector product of two vectors is also a vector.
- The angular momentum L of a particle is given by the vector product L = r × p, where p is its linear momentum.
- One can show that the vector product is distributive over addition but not associative, as evidenced by the Jacobi identity.
Learning
Memory Aids
Mnemonic
For a × b: Point index finger (a), middle finger (b), thumb shows result. Like an 'X' marks the spot for crossing vectors.
Conceptual Metaphor
A twisting generator (the product generates a new direction of twist/torque/rotation from two directions of action).
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Do not translate directly as "векторное произведение" and then assume it's the same as "скалярное произведение" (dot product). The Russian term is correct, but confusion arises between the two operations.
- The English "product" in maths covers both операция умножения (operation) and результат (result).
Common Mistakes
- Forgetting the right-hand rule for direction.
- Treating it as commutative.
- Confusing it with the dot product when finding magnitude.
- Writing a × b = |a||b| sin θ (missing the unit vector n̂).
Practice
Quiz
What is a key property of the vector product a × b?
FAQ
Frequently Asked Questions
Yes, in the context of three-dimensional Euclidean space, 'vector product' and 'cross product' are synonymous.
Not directly. The standard vector product is defined for two vectors in 3D space. In 2D, the result would be a scalar (the magnitude of the would-be 3D product along the perpendicular axis).
Because in general, a × (b × c) is not equal to (a × b) × c. This is why the order of operations in expressions involving multiple cross products is critical.
The vector product of parallel (or antiparallel) vectors is the zero vector, as the sine of the angle (0° or 180°) between them is zero.