Question about Morin's time dilation explanation

[robphy]

If a primed frame is in relative motion to an unprimed frame do you use the primed time to describe the motion or the unprimed time?

Quoting myself from another thread you started,

I suggest first looking at the Euclidean analogue: https://en.wikipedia.org/wiki/Rotation_matrix

Many questions and confusions in special relativity
can be addressed by first asking what happens in
hopefully simpler and more familiar analogues in Euclidean geometry or Galilean physics.

IMHO, https://www.merriam-webster.com/dictionary/walk before one can run

 

[Mister T]

If a primed frame is in relative motion to an unprimed frame do you use the primed time to describe the motion or the unprimed time?

The entire point of creating the primed and unprimed frames is that they are in relative motion. You use both of them! Otherwise there would be no reason to create both of them.

 

[Sagittarius A-Star]

The full Lorentz transform I find a little confusing, I don't fully understand the part where $\frac {v} {c^2} \Delta x'$ the units seem to check out but I'm not sure how that quantity is associated with time.

It's the term for "relativity of simultaneity". Such a term appeared also in your calculation:

...
$vt' = \gamma v ({t - x\frac {v}{c^2}})$

$t' = \gamma ({t - \frac {v}{c^2}}x)$

 

[Chenkel]

It's the term for "relativity of simultaneity". Such a term appeared also in your calculation:

I appreciate that you helped me get that result hopefully I see the big picture and things start making sense.

Thank you for the help.