Vector Product

Vector Product: Key Concepts

• Also known as the cross product, the vector product is an operation between two vectors in three-dimensional space.
• The resultant vector from the vector product is perpendicular to the plane formed by the original vectors.
• The vector product of two vectors a and b, denoted a × b, is defined as a vector whose magnitude is equal to the area of the parallelogram spanned by a and b, and its direction is given by the right-hand rule.
• The vector product is not commutative. In other words, a × b ≠ b × a, but instead a × b = - (b × a).

Calculation of Vector Product

• The vectors a and b with components a = (a1, a2, a3) and b = (b1, b2, b3) respectively, the vector product a × b is calculated using the determinant of a matrix formed by the vectors’ components: a × b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1).

• The magnitude of the cross product is given by the area of the parallelogram spanned by the original vectors, calculated as

  a × b

  =

  a

  b

  sin(θ), where θ is the angle between a and b.

• The right-hand rule is employed to determine the direction of the cross product. Curling the fingers of the right hand from a towards b, the thumb points in the direction of a × b.

Properties of the Vector Product

• The vector product is distribution over addition i.e. a × (b + c) = a × b + a × c.
• It is anticommutative, meaning that swapping the order of the vectors changes the sign of the result: a × b = - (b × a).
• It is not associative, that is, a × (b × c) ≠ (a × b) × c.
• Vector product of a vector with itself is always zero, as the angle between them is zero: a × a = 0.

Applications of Vector Product

• The vector product is used in understanding rotations, torque and angular momentum in physics, and modelling physical systems.
• Its properties also make it helpful in computer graphics and mechanics for finding perpendicular vectors and testing for parallelism and coplanarity.