# Generalized Galilean transformation

1. May 18, 2017

### Pushoam

1. The problem statement, all variables and given/known data
Write the Galilean coordinate transformation equations
for the case of an arbitrary direction for the relative velocity v of one frame with respect to the other. Assume that the corresponding axes of the two frames remain parallel. (Hint: let v have componentsvx, vy, vz.)

Write down the equivalent matrix equation.
2. Relevant equations
Consider a frame S' moving with uniform velocity v with respect to another inertial frame S.
Then
x'=x- vx t, y'=y- vy t, z'=z- vz t, t' = t

3. The attempt at a solution
The matrix formulation is
$$\begin{pmatrix} x' \\ y' \\ z'\\ t' \end{pmatrix} =\begin{pmatrix} x& -v_x \\ y & -v_y \\ z & -v_z\\ 0&1\end{pmatrix} \begin{pmatrix} 1 \\ t\\ \end{pmatrix}$$
Is this right?

2. May 18, 2017

### BvU

I don't think so. Your relevant equations are fine. For a coordinate transformation one expects something of the form $$\begin{pmatrix} x' \\ y' \\ z' \\ t' \end{pmatrix} = M\begin{pmatrix} x \\ y \\ z \\ t \end{pmatrix}$$
(compare with the matrix for a simple rotation in 3D, e.g. around the z-axis)

3. May 18, 2017

### Pushoam

Then,
$\begin{pmatrix} x' \\ y' \\ z'\\ t' \end{pmatrix} =\begin{pmatrix} 1&0&0& -v_x \\ 0&1&0& -v_y \\ 0&0&1 & -v_z\\ 0&0&0&1\end{pmatrix}\begin{pmatrix} x \\ y\\ z\\ t \end{pmatrix}$
Is this correct?
Can you please tell me how to write the matrices side by side?

Last edited: May 18, 2017
4. May 18, 2017

### BvU

I think this is what the exercise composer meant, yes.
And the inverse transformation matrix looks the same, except that the minus signs are now plus signs. Good exercise to check that $M^{-1}M = MM^{-1} = {\mathbb I}$
But you did that already in post #1 !

Generally: right-click on a formula and pick show math as $\TeX$ commands

5. May 18, 2017

### Pushoam

Thank you.
I got it. I need to write" ## "before and after the text command for matrices.