# hypothetical 4 dimensional vector space

Before I start asking the question, I want to apologize for my illiteracy in latex maths and the abstraction of my question.

My question is, is it possible for me to define a hypothetical 4 dimensional vector space? For example, if I defined a b c and d as my four unit vectors, can I create an orthogonal vector space where they are all orthogonal to each other. It is not possible spatially I can understand because spatial cartesian system is limited to 3 unit vectors and the 4th must be a linear combination of the other 3, but is it mathematically possible to have an orthogonal vector space in 4 dimensions?? I don't want to visualize it, but does mathematics allow for it? And how will my vector products of my space be defined in that situation? I understand that my inner products of my unit vectors with each other will be zero.

• There are very real (mathematically real, I mean) four, or any number, dimensional vector spaces. – Timbuc Jan 10 '15 at 10:41
• This is routine in mathematics. And no, cartesian systems are not limited to 3 directions. en.wikipedia.org/wiki/… – Raskolnikov Jan 10 '15 at 10:43
• Mathematicians are brave and strong. They could handle any dimensional space. To infinity and beyond. – sas Jan 10 '15 at 11:07

Mathematically there is no reason to limit ourselves to three dimensions. To have a four-dimensional vector space, we need four components for our vectors. We can use $\{(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1) \}$ as an orthonormal basis for our space. This space is generally denoted $\mathbb{R}^4$.
Yes this is possible. Simply let $a,b,c,d$ be your unit vectors and define the inner product of $a,b,c,d$ to be $0$ ( so $(a,b) = 0$ etc., but of course we still need $(a,a) = 1$). The inner product of all the other vectors in the space, which are given by $$v= \lambda_a a +\lambda_b b +\lambda_c c + \lambda_d d,$$ can now be found by using the linearity of the inner product.
• Crossproduct can be generalised. The best way seems to be multiplying $d-2$ vectors. – BartekChom Jan 10 '15 at 13:25