Editing Euclidean vector (section)

===Other dimensions===
With the exception of the cross and triple products, the above formulae generalise to two dimensions and higher dimensions. For example, addition generalises to two dimensions as 
<math display=block>(a_1{\mathbf e}_1 + a_2{\mathbf e}_2)+(b_1{\mathbf e}_1 + b_2{\mathbf e}_2) = (a_1+b_1){\mathbf e}_1 + (a_2+b_2){\mathbf e}_2,</math>
and in four dimensions as
<math display=block>\begin{align}
(a_1{\mathbf e}_1 + a_2{\mathbf e}_2 + a_3{\mathbf e}_3 + a_4{\mathbf e}_4) &+ (b_1{\mathbf e}_1 + b_2{\mathbf e}_2 + b_3{\mathbf e}_3 + b_4{\mathbf e}_4) =\\
(a_1+b_1){\mathbf e}_1 + (a_2+b_2){\mathbf e}_2 &+ (a_3+b_3){\mathbf e}_3 + (a_4+b_4){\mathbf e}_4.
\end{align}</math>

The cross product does not readily generalise to other dimensions, though the closely related [[Exterior algebra#Areas in the plane|exterior product]] does, whose result is a [[bivector]]. In two dimensions this is simply a [[pseudoscalar]]
<math display=block>(a_1{\mathbf e}_1 + a_2{\mathbf e}_2)\wedge(b_1{\mathbf e}_1 + b_2{\mathbf e}_2) = (a_1 b_2 - a_2 b_1)\mathbf{e}_1 \mathbf{e}_2.</math>

A [[seven-dimensional cross product]] is similar to the cross product in that its result is a vector orthogonal to the two arguments; there is however no natural way of selecting one of the possible such products.