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The magnitude of an n-dimensional vector $\langle x_1, x_2, x_3, \dots, x_n\rangle$ can be written as:

$magnitude(n) = \sqrt[2]{\sum_{i=1}^n \Delta x_i^2}$

The number $2$ appears twice in this equation. Why does it appear at all, and why isn't it some other number?

Thomas Andrews
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Ganymede
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    Basically, by Pythagoras, this is how distance is measured. – Randall Sep 19 '23 at 20:29
  • See https://math.stackexchange.com/questions/221367/why-do-we-use-the-euclidean-metric-on-mathbbr2 or https://math.stackexchange.com/questions/452836/why-is-the-2-norm-special?noredirect=1&lq=1 – Dennis Sep 19 '23 at 20:30
  • @Dennis thanks but neither of those answer my question directly – Ganymede Sep 19 '23 at 20:32
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    It is nothing but an extension of the Pythagorean theorem. Considering the horizontal and vertical components of a 2d vector, the length of the vector can be thought of as the hypotenuse of a right-angled triangle whose shorter legs are equal to the components of said vector. Thus the length of the vector is equal to the square root of the sum of the squares of the components. One simply adds more components for higher-dimensional vectors – H. sapiens rex Sep 19 '23 at 20:32
  • Yes, that answers it @Jan And everyone else above. I appreciate it. – Ganymede Sep 19 '23 at 20:34
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    It is not the magnitude of $n,$ but of something else: $\Delta x=\langle \Delta x_i\rangle_{i=1}^n.$ – Thomas Andrews Sep 19 '23 at 20:36
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    Note that it is possible to use some other number instead of 2, giving a p-norm. However, only $p = 2$ gives the conventional geometric length of the vector. – Dan Sep 19 '23 at 20:37

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