Euclidean norm vs pythagorean theorem?I recently stumbled upon an Euclidian norm. First I thought there are the powers and square root to deal with possible negative values (like in Standard deviation formula) but then I realized, the final number (sum of squares and square root of it) is not same as sum of absolute numbers.So then I noticed it is more related to Pythagorean theorem (Euclidian distance).But if looked in google for PT in 3d space i found the formula like ( a 2 + b 2 + c 2 ) 1 2 But in the Euclidian norm it is S Q R T ( a 2 + b 2 + c 2 )How comes, it is different?Second question, how comes that the theorem works for any count of dimensions?

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Euclidean norm vs pythagorean theorem?I recently stumbled upon an Euclidian norm. First I thought there are the powers and square root to deal with possible negative values (like in Standard deviation formula) but then I realized, the final number (sum of squares and square root of it) is not same as sum of absolute numbers.So then I noticed it is more related to Pythagorean theorem (Euclidian distance).But if looked in google for PT in 3d space i found the formula like  (      a    2    +      b    2    +      c    2        )          1      2      But in the Euclidian norm it is  S  Q  R  T  (      a    2    +      b    2    +      c    2    )How comes, it is different?Second question, how comes that the theorem works for any count of dimensions?

Alisa Gilmore · Accepted Answer

The two formulas are indeed the same.You can generalize it to n dimensions by repeated application of Pythagoras:  ‖  (  a  ,  b  ,  c  )  ‖  =  ‖  (  a  ,  ‖  (  b  ,  c  )  ‖  )  ‖  =  ‖  (  a  ,            b      2        +          c      2        )  ‖  =            a      2        +    (          b      2        +          c      2        )  and so on.(In this reasoning, you project (a,b,c) to the plane   y  z, giving (b,c) as an intermediate point, joining the origin to (b,c), then (b,c) to (a,b,c).)

Euclidean norm vs pythagorean theorem? I recently stumbled upon an Euclidian norm. First I thought

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