Why Are the Terms Squared in the Lorentz Transformation?

Dark_knight90
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Hello
This is a part of a simple paper about special relativity

[PLAIN]http://img15.imageshack.us/img15/8789/91001769.jpg

I don't understand the assumption in the red box .. why are they all squared ?

thank you
 
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Do you remember the Pythagorean rule? In two-dimensional space, \Delta x and \Delta y are the lengths of two sides of a right triangle, and \Delta r is the length of the hypotenuse:

(\Delta r)^2 = (\Delta x)^2 + (\Delta y)^2

What you have is the three-dimensional version.
 
That's basically an extension of the Pythagorean theorem which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two sides. You can then repeat the process, adding in the square of the length of the third side giving you the square of the total length.
 
Got it .. Thank you :)
 
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