How is the area of a parallelogram equal to the length of a vector. When we take the cross product between two vectors we get another vector with length equal to the area of the parallelogram spanned by the aforementioned vectors. How can area be equaled to length?
 A: The assertion "If $u$ and $v$ are vectors in $\mathbf{R}^{3}$, then $\|u \times v\|$ is equal to the area of the parallelogram spanned by $u$ and $v$" presumes you're in Euclidean geometry with a "standard basis", i.e., an ordered set of three mutually-orthogonal unit vectors. In this setting:


*

*The magnitude of a vector can be associated with a real number, the magnitude of a standard basis vector serving as "unit";

*The area of a plane polygon can be associated with a real number, the area of a square spanned by a pair of standard basis vectors serving as "unit";

*The volume of a polyhedron can be associated with a real number, the volume of the cube spanned by the standard basis vectors serving as "unit".
If $u = (u_{1}, u_{2}, u_{3})$ and $v = (v_{1}, v_{2}, v_{3})$, the numerical area of the parallelogram spanned by $u$ and $v$ is
$$
\sqrt{(u_{2}v_{3} - u_{3}v_{2})^{2} + (u_{3}v_{1} - u_{1}v_{3})^{2} + (u_{1}v_{2} - u_{2}v_{1})^{2}}.
$$
This happens to be the numerical magnitude of the vector $u \times v$.
There's underlying mathematical machinery lurking. The Wikipedia articles on exterior algebra and the Hodge $*$-operator may be of interest.
