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

glitinosim3

glitinosim3

Answered

2022-07-14

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?

Answer & Explanation

persstemc1

persstemc1

Expert

2022-07-15Added 18 answers

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 "geometrically", 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 ...

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