Further Vectors: Vector product

Further Vectors: Vector product

Understanding the Vector Product

  • The vector product is a binary operation that combines two vectors to produce a third vector. This product is also known as the cross product.

  • When two vectors 𝐚 and 𝐛 are crossed, the result is another vector 𝐜 = 𝐚 × 𝐛. This vector is perpendicular to the plane containing 𝐚 and 𝐛.

  • The magnitude of vector 𝐜 is equal to the product of the magnitudes of 𝐚 and 𝐛 and the sine of the angle between them, denoted as 𝐚 . 𝐛 .sin𝜃.
  • The direction of vector 𝐜 can be determined using the right-hand rule. If the fingers of the right hand are curled from 𝐚 to 𝐛, the thumb will point in the direction of 𝐚 × 𝐛.

Properties of the Vector Product

  • The vector product is not commutative, which means 𝐚 × 𝐛 ≠ 𝐛 × 𝐚. In fact, these two vectors are equal in magnitude but opposite in direction, so 𝐚 × 𝐛 = -𝐛 × 𝐚.

  • The vector product is distributive over addition. For any three vectors 𝐚, 𝐛, 𝐜, it is true that 𝐚 × (𝐛 + 𝐜) = 𝐚 × 𝐛 + 𝐚 × 𝐜.

  • The vector product of a vector with itself yields the zero vector, i.e., 𝐚 × 𝐚 = 0.

  • The vector product is antisymmetric, which means for any vectors 𝐚 and 𝐛, if 𝐚 × 𝐛 = 0 then 𝐚 and 𝐛 are either the zero vector or they are parallel.

Practical Applications of the Vector Product

  • The vector product is used to find a normal (perpendicular) vector to two given vectors, useful in geometry and physics.

  • It also helps calculate the area of a parallelogram spanned by two vectors; the area = 𝐚 × 𝐛 .
  • Understanding the vector product is key to explore complex concepts in maths and physics, including electromagnetic fields, angular momentum, and torque.