When using the definition and properties of the inner product, we get the parallelogram law: ‖ x + y ‖ 2 = ⟨ x + y , x + y ⟩ = ⟨ x , x ⟩ + ⟨ x , y ⟩ + ⟨ y , x ⟩ + ⟨ y , y ⟩ , ‖ x − y ‖ 2 = ⟨ x − y , x − y ⟩ = ⟨ x , x ⟩ − ⟨ x , y ⟩ − ⟨ y , x ⟩ + ⟨ y , y ⟩Adding these two expressions: ‖ x + y ‖ 2 + ‖ x − y ‖ 2 = 2 ⟨ x , x ⟩ + 2 ⟨ y , y ⟩ = 2 ‖ x ‖ 2 + 2 ‖ y ‖ 2 as required. But the above does not simplify to Pythagoras' theorem ‖ x + y ‖ 2 = ‖ x ‖ 2 + ‖ y ‖ 2 if x and y are orthogonal.How can we get Pythagoras from the parallelogram law?

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When using the definition and properties of the inner product, we get the parallelogram law:      ‖  x  +  y      ‖    2    =    ⟨  x  +  y  ,  x  +  y  ⟩  =  ⟨  x  ,  x  ⟩  +  ⟨  x  ,  y  ⟩  +  ⟨  y  ,  x  ⟩  +  ⟨  y  ,  y  ⟩    ,    ‖  x  −  y      ‖    2    =    ⟨  x  −  y  ,  x  −  y  ⟩  =  ⟨  x  ,  x  ⟩  −  ⟨  x  ,  y  ⟩  −  ⟨  y  ,  x  ⟩  +  ⟨  y  ,  y  ⟩Adding these two expressions:  ‖  x  +  y      ‖    2    +  ‖  x  −  y      ‖    2      =    2  ⟨  x  ,  x  ⟩  +  2  ⟨  y  ,  y  ⟩    =    2  ‖  x      ‖    2    +  2  ‖  y      ‖    2  as required. But the above does not simplify to Pythagoras&#039; theorem   ‖  x  +  y      ‖    2    =  ‖  x      ‖    2    +  ‖  y      ‖    2  if   x and   y are orthogonal.How can we get Pythagoras from the parallelogram law?

persstemc1 · Accepted Answer

If you are talking strictly about the parallelogram, when the vector are orthogonal (hence   ⟨  x  ,  y  ⟩  =  0, you have      |        |    x  +  y      |              |        2    =  ⟨  x  +  y  ,  x  +  y  ⟩  =  ⟨  x  ,  x  ⟩  +  ⟨  x  ,  y  ⟩  +  ⟨  y  ,  x  ⟩  +  ⟨  y  ,  y  ⟩  =  ‖  x      ‖    2    +  ‖  y      ‖    2  You can also see it &quot;geometrically&quot;, by noticing that this means that when   x and   y are orthogonal, you have a rectangle, with both diagonals of same length, that is      |        |    x  +  y      |              |        2    =      |        |    x  −  y      |              |        2  ...

When using the definition and properties of the inner product, we get the parallelogram law:

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