Not sure if this is geometry or Algebra, but I was wondering, why the axes for cartesian representations are at right angles to each other?
Because orthogonality is useful as a basis for the coordinate system. In higher mathematics, you will find that given any arbitrary representation for a point in Euclidean space, there exists an orthogonal basis which can be used to represent it. This is a powerful result in linear algebra. There exist other coordinate systems such as projective geometries where orthogonality works differently than what you expect. For example, a line can intersect itself in a projective space, and two orthogonal lines can intersect more than once.
A major reason for using orthonormal coordinates is convenience. When writing v in terms of an orthonormal basis, $\displaystyle \{e_1, e_2, ..., e_n\}$, the coefficient of $\displaystyle e_i$ is the dot product of v and $\displaystyle e_i$.