Vector product

Definition

The vector product, also known as the cross product, is a binary operation on two vectors in three-dimensional space. It results in a third vector that is perpendicular to the plane of the original vectors and has a magnitude equal to the area of the parallelogram they span.

5 Must Know Facts For Your Next Test

1. The vector product of two vectors $\mathbf{A}$ and $\mathbf{B}$ is denoted by $\mathbf{A} \times \mathbf{B}$.
2. The direction of the resulting vector from a cross product is determined by the right-hand rule.
3. The magnitude of the cross product $|\mathbf{A} \times \mathbf{B}|$ is given by $|\mathbf{A}| |\mathbf{B}| \sin(\theta)$, where $\theta$ is the angle between $\mathbf{A}$ and $\mathbf{B}$.
4. The cross product is anti-commutative, meaning $\mathbf{A} \times \mathbf{B} = - (\mathbf{B} \times \mathbf{A})$.
5. The result of a cross product operation is always orthogonal to both original vectors.

Review Questions

• What rule determines the direction of the resulting vector in a vector product?
• How can you calculate the magnitude of a cross product?
• What does it mean for a cross product to be anti-commutative?

Related terms

Dot Product:

The dot product, or scalar product, is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by performing pairwise multiplications and summing them up.

Right-Hand Rule:

A mnemonic used to determine the direction of the resultant vector in a cross-product operation: point your right-hand fingers toward the first vector and curl them towards the second; your thumb points in the direction of the resultant.

Orthogonal Vectors: Vectors are orthogonal if their dot product equals zero, indicating that they are perpendicular to each other.