cross product

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cross product

A student calculates the cross product of two vectors on a chalkboard.

Definition
  1. Noun:
    • A vector resulting from a specific binary operation on two vectors in three-dimensional space: The cross product is a mathematical operation that takes two vectors and produces a third vector that is perpendicular to the plane containing the original two vectors. Its magnitude is equal to the product of the magnitudes of the two vectors and the sine of the angle between them.
Usage Examples
  • Noun:
    • The cross product of two vectors is essential for calculating torque in physics.
    • To find a vector normal to a surface, you can compute the cross product of two tangent vectors.
    • The result of the cross product is a pseudovector, meaning its direction depends on the handedness of the coordinate system.
Advanced Usage

  • "Right-hand rule": A common mnemonic for determining the direction of the resulting cross product vector. If you point your index finger in the direction of the first vector and your middle finger in the direction of the second vector, your thumb will point in the direction of their cross product.

    • Remember to use the right-hand rule to check the direction of the cross product.

  • "Zero cross product": The cross product of two parallel or anti-parallel vectors is the zero vector.

    • If two vectors are collinear, their cross product will be zero.
Variants and Related Words

  • Vector product (n): A synonym for cross product.

    • The terms 'cross product' and 'vector product' are often used interchangeably.

  • Dot product (n): A different, scalar-producing operation on two vectors.

    • Unlike the cross product, the dot product of two vectors results in a scalar quantity.

  • Scalar triple product (n): A scalar value obtained from the dot product of one vector with the cross product of two others.

    • The scalar triple product gives the volume of the parallelepiped formed by three vectors.
Synonyms
  • Outer product (in certain contexts, though this can have a broader meaning in tensor algebra).
  • Vector product.
Related Phrases (Phrasal Verbs)

(Not applicable for this mathematical term.)

Related Idioms

(Not applicable for this mathematical term.)

cross product

A student calculates the cross product of two vectors on a chalkboard.

Noun
  1. a vector that is the product of two other vectors

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