TITLE: How is "area" a vector?
QUESTION [10 upvotes]: "We consider Area as a vector."

How is an area a vector? Why is that the vector is always normal to the area element?

REPLY [3 votes]: Actually, you can have a n-dimensional cross-product
which generalizes the 3-dimensional cross-product.
If $A_i, i=1, n-1$ 
are $n-1$ $n$-dimensional vectors
with the elements of $A_i$
being
$a_{i,1}, a_{1,2}, ..., a_{i, n}$,
and the $n$ unit vectors are
$e_1, ..., e_n$,
then the following determinant 
is a n-dimensional vector orthogonal to
all the $A_i$:
$\left|
\begin{array}{cccc}
e_1 & e_2 & ... & e_n \\
A_{1,1} & A_{1,2} & ... & A_{1,n} \\
A_{2,1} & A_{2,2} & ... & A_{2,n} \\
... & ... & ... & ... \\
A_{k,1} & A_{k,2} & ... & A_{k,n} \\
... & ... & ... & ... \\
A_{n-1,1} & A_{n-1,2} & ... & A_{n-1,n} \\
\end{array} \right|$