Vector product

Vector Product

  • The vector product, also known as the cross product, is an operation that takes two vectors and returns a vector which is orthogonal (perpendicular) to both of them.

  • If you have two vectors a and b, their vector product is defined as **a x b = a   b sinθ n**, where ** a ** and ** b ** are the magnitudes of a and b, θ is the angle between a and b, and n is a unit vector perpendicular to the plane containing a and b.
  • The magnitude of the vector product is equal to the area of the parallelogram that the vectors span.

  • Importantly, the vector product is not commutative, meaning that a x b ≠ b x a. In fact, a x b = - b x a. This follows from the right-hand rule, which determines the direction of the resultant vector.

Right-Hand Rule

  • The right-hand rule is a way to remember the direction of the cross product vector. Point your index finger in the direction of the first vector (a), then point your middle finger in the direction of the second vector (b). Your thumb will point in the direction of a x b.

  • If the cross-product vector points in the opposite direction, the order of the vectors is reversed.

Properties of the Vector Product

  • The vector product satisfies the distributive law, i.e. a x (b + c) = a x b + a x c.

  • However, the vector product does not satisfy the commutative law, i.e. a x b ≠ b x a but a x b = -b x a.

  • The vector product does satisfy the associative law, i.e. a x (b x c) = (a x b) x c.

  • If two vectors are parallel, their vector product is the zero vector. This follows from the fact that sin(0°) = 0, and the definition of the cross product involves the product of magnitudes and sin(θ).

Applications of Vector Product

  • The vector product can be used to find a vector that is perpendicular to a plane defined by two other vectors.

  • This operation plays a crucial role in geometrical applications such as calculating the area of triangle and parallelogram formed by two vectors.

  • In physics, the vector product is often used to calculate torque and angular momentum.