How to Calculate the Area Between Two Vectors in R^n?

[poissonspot]

Hi, I wondered whether a well known expression is known that computes the area between two vectors in R^n. By area between two vectors, I mean the area that would be computed by considering the subspace spanned by the two, projecting the entire space to a "parallel plane" and then finally given by the magnitude of the cross product of the image of the vectors under this projection. Thank you,

 

[Opalg]

Hi, I wondered whether a well known expression is known that computes the area between two vectors in R^n. By area between two vectors, I mean the area that would be computed by considering the subspace spanned by the two, projecting the entire space to a "parallel plane" and then finally given by the magnitude of the cross product of the image of the vectors under this projection.

In other words, given vectors $a,b$ in $\mathbb{R}^n$, you want the area $A$ of the parallelogram with vertices at the origin, $a$, $b$ and $a+b.$ If the angle between the vectors is $\theta$, this can be expressed in terms of the dot product by $$A = |a||b|\sin\theta = \sqrt{|a|^2|b|^2(1-\cos^2\theta)} = \sqrt{|a|^2|b|^2 - (a\cdot b)^2}.$$