Unit content
Cross product
In three-dimensional space, the cross product of two vectors produces a third vector perpendicular to both:
$$\mathbf u\times\mathbf v.$$
Unlike the dot product, which produces a scalar, the cross product produces a vector.
Direction
The direction of $\mathbf u\times\mathbf v$ is perpendicular to the plane containing $\mathbf u$ and $\mathbf v$. Its orientation follows the right-hand rule. Reversing the order reverses the result:
$$\mathbf v\times\mathbf u=-(\mathbf u\times\mathbf v).$$
In particular,
$$\mathbf v\times\mathbf v=\mathbf0.$$
Magnitude and area
If $\theta$ is the angle between the vectors, then
$$|\mathbf u\times\mathbf v| =|\mathbf u|,|\mathbf v|\sin\theta.$$
This is exactly the area of the parallelogram spanned by $\mathbf u$ and $\mathbf v$. Parallel vectors have zero cross product because the parallelogram collapses to zero area.
Components
For
$$\mathbf u=(u_x,u_y,u_z),\qquad \mathbf v=(v_x,v_y,v_z),$$
the cross product is
$$\mathbf u\times\mathbf v= (u_yv_z-u_zv_y,;u_zv_x-u_xv_z,;u_xv_y-u_yv_x).$$
The cross product is specific to the geometry of three-dimensional vectors in this curriculum and later appears naturally in torque, magnetic force and oriented surface calculations.