Definition

1. Noun:
   • A vector resulting from a specific binary operation on two other vectors in three-dimensional space: The vector product, also known as the cross product, is a vector quantity. Its magnitude is equal to the product of the magnitudes of the two original vectors and the sine of the angle between them. Its direction is perpendicular to the plane containing the two original vectors, following the right-hand rule.

Usage Examples

• Noun:
  • The vector product of force and displacement gives the torque.
  • To find a normal to the surface, calculate the vector product of the two tangent vectors.
  • In physics, angular momentum is defined as the vector product of the position vector and the linear momentum vector.

Advanced Usage

• Mathematical Notation: The vector product of vectors a and b is denoted as a × b.
  • The result of the calculation a × b is itself a vector.
• Geometric Interpretation: The magnitude of the vector product represents the area of the parallelogram spanned by the two original vectors.
  • The magnitude of the vector product |a × b| equals the area of the parallelogram with sides a and b.

Variants and Related Words

• Cross Product (n): A complete synonym for vector product.
  • The terms 'cross product' and 'vector product' are used interchangeably.
• Scalar Product (n): A related but distinct operation (the dot product), which results in a scalar, not a vector.
  • Unlike the vector product, the scalar product of two vectors yields a single number.

Synonyms

• Cross Product: The most common direct synonym.
• Outer Product: This term can sometimes refer to the vector product, though it has broader meanings in advanced algebra.

Related Phrases

• Right-Hand Rule: A mnemonic for determining the direction of the resulting vector product.
  • Apply the right-hand rule to find the direction of the vector product.
• Perpendicular Vector: Describes the directional property of the result.
  • The vector product always yields a vector perpendicular to the plane of the two input vectors.

Noun

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