Cross product

Definition

The cross product is a binary operation on two vectors in three-dimensional space, resulting in another vector that is perpendicular to the plane containing the original vectors. It is denoted by $\mathbf{A} \times \mathbf{B}$ and has both magnitude and direction.

5 Must Know Facts For Your Next Test

1. 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 $\mathbf{B}$.
2. The direction of the cross product follows the right-hand rule: if you point your index finger in the direction of $\mathbf{A}$ and your middle finger in the direction of $\mathbf{B}$, your thumb points in the direction of $\mathbf{A} \times \mathbf{B}$.
3. If two vectors are parallel or anti-parallel, their cross product is zero because $\sin(0) = 0$ or $\sin(180^\circ) = 0$.
4. In component form, for vectors $\mathbf{A} = (A_x, A_y, A_z)$ and $\mathbf{B} = (B_x, B_y, B_z)$, the cross product is computed as $(A_y B_z - A_z B_y, A_z B_x - A_x B_z, A_x B_y - A_y B_x)$.
5. The cross product is antisymmetric: swapping the order of multiplication changes its sign ($\mathbf{A} \times \mathbf{B} = - (\mathbf{B} \times \mathbf{A})$).

Review Questions

• What physical quantity can be represented by the magnitude of a cross product?
• How do you determine the direction of a vector resulting from a cross product?
• What happens to the result of a cross product if two vectors are parallel?

Related terms

Dot Product: An operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and summing those products.

Right-Hand Rule: A convention used to determine the direction of certain vector quantities such as angular velocity or magnetic field relative to current flow; place your right hand with fingers aligned with one vector and rotate towards another.

Vector Components: Projections of a vector along coordinate axes; any vector can be expressed as sums of its components along each axis.