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.
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$\begingroup$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 $\mathbb{R}^4$.
$\endgroup$ $\begingroup$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.
If you also insist on defining a crossproduct, things get much more complicated and I don't even know whether this is definable in general: this really depends on what you think is the fundamental definition of the cross product (its meaning as the area of a parallellogram,as a given function of two vectors etc.).
In general, it is possible to define this for any number of basis vectors. To add to the complexity, it is even possible to define other inner products (such as the Minkowski inner product used in Special and General Relativity) and to allow for an infinite number of dimensions. The vectors can be arrows, but really vectors can be anything that allow for scalar multiplication and addition, such as functions, matrices etc. Vector spaces have lots of applications!
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