The cross product between two vectors in the three-dimensional space is a fundamental operation in multivariable calculus. It is used to construct many vector-field operations. Note that it is only defined in the three-dimensional space.
For two vectors \(\vec{u}\) and \(\vec{v}\) in the three-dimensional space, their cross product is denoted by \(\vec{u} \times \vec{v}\). It is itself a vector in the three-dimensional space that is perpendicular to both \(\vec{u}\) and \(\vec{v}\). More precisely, it is defined by the formula:
\[ \vec{u} \times \vec{v} = \| \vec{u} \| \| \vec{v} \| \sin(\theta) \vec{n} \]
where \(\theta\) is the angle between \(\vec{u}\) and \(\vec{v}\) and \(\vec{n}\) is the unit vector given by the right-hand rule.
It is important to remember that the cross product is not commutative. Indeed, it is anti-commutative in the sense that \(\vec{v} \times \vec{u} = - (\vec{u} \times \vec{v})\). This tells us that the cross product also encodes the information about the orientation.
In the Cartesian coordinate system, we can also compute the cross product using a (formal) 3x3 matrix determinant.
\[ \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix} \times \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix} = \det \begin{bmatrix} \vec{i} & \vec{j} & \vec{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{bmatrix} \]
Here, \(\vec{i},\vec{j},\vec{k}\) are the standard basis vectors (unit vectors pointing in the \(x,y,z\) directions respectively).
Using a basic trigonometry, we can verify that the magnitude of the cross product of two vectors is precisely the area of the parallelogram spanned by the two vectors (homework).