College Physics II – Mechanics, Sound, Oscillations, and Waves
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.
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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 $\mathbf{B}$.
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 $\mathbf{A} \times \mathbf{B}$.
If two vectors are parallel or anti-parallel, their cross product is zero because $\sin(0) = 0$ or $\sin(180^\circ) = 0$.
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 $(A_y B_z - A_z B_y, A_z B_x - A_x B_z, A_x B_y - A_y B_x)$.
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?
An operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and summing those products.
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.