Why do I , J and K in mechanics represent X , Y and Z in maths?

Question

Why are letters $i$, $j$, and $k$ used for axes names in mechanics while letters $x$ , $y$ and $z$ are used in mathematics?

Why these dimensions weren't called A, B and C or F, G and H?

Answer 1

This usage of $\mathbf i$, $\mathbf j$, and $\mathbf k$ is not specific to physics. It is also used in mathematics, specifically when teaching linear algebra or multivariable calculus in $\mathbf R^3$ as well as group theory for the quaternion group $Q_8 = \{\pm 1, \pm i, \pm j, \pm k\}$.

The notation came from the use of the letters $i$, $j$, and $k$ in quaternions, which were created by Hamilton (a physicist). They are a basis of the 3-dimensional space of "pure quaternions" or "imaginary quaternions" $\{bi + cj + dk : b, c, d, \in \mathbf R\}$, and conjugation on that subspace by all nonzero quaternions (or just by the unit sphere $S^3$) provides a quaternionic description of proper rotations in 3-dimensional space. This made the space of pure quaternions a natural model of physical 3-dimensional space on which you could not only add elements but also describe their rotations in a novel (at the time) way through conjugation by a 4-dimensional quaternion.

Hamilton called the pure quaternion part of a quaternion $a+bi+cj+dk$ a “vector” and the real part of a quaternion a "scalar". This was in the 1840s, before linear algebra was well-developed, and this usage by Hamilton is how we got the labels "vector" and “scalar” in linear algebra. Search for vector on this page.

The first group theory paper that mentions the quaternion group is Cayley's 1859 paper, where he writes the elements as
$$ 1, \vartheta, \alpha, \beta, \gamma, \vartheta\alpha, \vartheta\beta, \vartheta\gamma. $$ He says it looks simpler if he writes $\vartheta$ as $-1$ and that $\alpha, \beta$, and $\gamma$ "combine according to the laws of the quaternion symbols $i$, $j$, $k$."

In Dedekind's 1897 paper on nonabelian finite groups in which every subgroup is normal ($Q_8$ is the first example), Dedekind did not use the $i, j, k$ notation. He called the group $Q$, not $Q_8$, and wrote $\alpha$, $\beta$, and $\gamma$ instead of $i$, $j$, and $k$, and he wrote $\varepsilon$ instead of $-1$: $Q = \{1, \varepsilon, \alpha, \varepsilon\alpha, \beta, \varepsilon\beta, \gamma, \varepsilon\gamma\}$.

Answer 2

A reference to deep pre-history and quaternions is interesting but it can be very misleading to imply the present tense. Quaternions are not used in anything remotely contemporary - math or physics. They have some residual use in game development but much more in pseudo-theoretic quibbles (to claim that Cartesian coordinates have intrinsic problems - problems are lazy Devs that change coordinates partially instead of calculating at least good, if not optimal paths for camera motions).

Historically speaking, Hamilton's quaternions stifled development of physics for about 50 years, until Gibbs introduced normal vector spaces and accompanying notation and then Heaviside used it to rewrite Maxwell's electrodynamics equations in a normal, vector form (Maxwell was a staunch quaternion supporter, that's how influential Hamilton was). Maxwell is so highly regarded as superior mathematician among physicists that to this day there are claims that he's the one who was "meant" to write Relativity and even Quantum Field Theory (he essentially invented the concept of fields) if he didn't die so early (at 48). One thing that gets easily forgotten is that he was completely stuck with quaternions.

In today's books $\mathbf i$, $\mathbf j$, $\mathbf k$ by and large denote just unit vectors $\hat{\mathbf{x}}$, $\hat{\mathbf{y}}$, $\hat{\mathbf{z}}$ or in a more general form $e_\mathbf{x}$, $e_\mathbf{y}$, $e_\mathbf{z}$ and then $e_\mathbf{1}$, $e_\mathbf{2}$, $e_\mathbf{3}$ or $e^\mathbf{1}$, $e^\mathbf{2}$, $e^\mathbf{3}$ to make it clear very early that there's more to come - N coordinates, no particular names, N getting big, going to infinity when it's convenient ... To that extent that might have been the reason why someone used i,j,k as well - to nudge the reader to detach conceptually from the all too familiar symbols. In $\mathbf R^3$ you'll have to go to $\boldsymbol{\hat{\mathbf{r}}}$, $\boldsymbol{\hat{\mathbf{\phi}}}$, $\boldsymbol{\hat{\mathbf{\theta}}}$, or $\boldsymbol{\hat{\mathbf{\rho}}}$, pretty much on the next page and latter you may end up with something like $\mathbf{q_\mathbf{i}}$, and then $q_{i}$ where it's assumed that unit vectors follow the names which just denote n-th coordinate.

In game development you are usually in $\mathbf R^4$ and the 4-th is usually denoted by $\mathbf{w}$ thereby $\hat{\mathbf{w}}$ - for no particular reason - just to have a different symbol for a coordinate that doesn't automatically mean that you have a Minkowski space of any form (the one used in Special Relativity has a very particular metric). Minkowski himself had a variant in which the 4-th coordinate was simply named $\mathbf t$. So the general lesson from history is - don't get attached to the names of coordinates.