Invariance of length [closed]

Invariance of interval in Minkowski space under coordinate transformation was proved by the postulates of special relativity. (https://physics.stackexchange.com/a/453536/213658 .see this answer) Is there any theory or postulates which proves that length is an invariant quantity in Euclidean space?

closed as off-topic by Aaron Stevens, Kyle Kanos, Jon Custer, ZeroTheHero, user191954 Feb 20 at 8:03

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• What do you mean by "in Euclidean space". Under what transformation are you referring to? – Aaron Stevens Feb 18 at 2:53
• Under Galilean transformation. – walber97 Feb 18 at 2:58
• Well have you tried to do it on your own? – Aaron Stevens Feb 18 at 3:00
• I did that using Galilean transformation. But I want to know is there any theory which proves the result Like the invariant interval I mentioned in the link. – walber97 Feb 18 at 3:02
• I'm not sure what you mean by "theory that proves the result". Just write out the length in the transformed coordinates and see that you get the same length as in the original coordinates. – Aaron Stevens Feb 18 at 3:04

I'm not sure I fully understand you still, but the Galilean transformation is (for transformation in one dimension, I'll leave multiple dimensions to you): $$x'=x-vt$$ $$y'=y$$ $$z'=z$$ $$t'=t$$
So the spatial length between two points $$(x_1',y_1',z_1')$$ and $$(x_2',y_2',z_2')$$ is given by $$L=\sqrt{(x_2'-x_1')^2+(y_2'-y_1')^2+(z_2'-z_1')^2}$$ Applying the transformation rules above: $$L=\sqrt{((x_2-vt)-(x_1-vt))^2+(y_2-y_1)^2+(z_2-z_1)^2}$$ $$L=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$$