SOLUTION: Show that 4u · v = ||u + v||^2 − ||u − v||^2. Show that u and v is orthogonal if and only if ||u + v|| = ||u − v||.

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Question 1203665: Show that 4u · v = ||u + v||^2 − ||u − v||^2. Show that u and v is orthogonal if and
only if ||u + v|| = ||u − v||.
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
Show that


4u+%2A+v+=+abs%28abs%28u+%2B+v%29%29%5E2+%E2%88%92+abs%28abs%28u+%E2%88%92+v%29%29%5E2

Hint. abs%28abs%28u%29%29%5E2=u%2Au, so

%28u%2Bv%29%5E2-%28u-v%29%5E2
=%28u%2Bv%29%28u%2Bv%29-%28u-v%29%28u-v%29
=u%5E2+%2B+2uv+%2B+v%5E2-%28u%5E2+-+2uv+%2B+v%5E2%29
=u%5E2+%2B+2uv+%2B+v%5E2-u%5E2+%2B2uv+-+v%5E2
=cross%28u%5E2%29+%2B+2uv+%2B+cross%28v%5E2%29-cross%28u%5E2%29+%2B2uv+-+cross%28v%5E2%29
= 2uv++%2B2uv+
= 4uv+



Show that u and v is orthogonal if and only if
abs%28abs%28u+%2B+v%29%29+=+abs%28abs%28u-v%29%29

If u and v are orthogonal vectors, then:

abs%28abs%28u%2Bv%29%29%5E2=abs%28abs%28u%29%29%5E2%2Babs%28abs%28v%29%29%5E2


now abs%28abs%28u%2Bv%29%29%5E2=abs%28abs%28u-v%29%29%5E2, but the norm is ever positive therefore:

abs%28abs%28u%2Bv%29%29=abs%28abs%28u-v%29%29
.
now, if abs%28abs%28u%2Bv%29%29=abs%28abs%28u-v%29%29 we have:
abs%28abs%28u%2Bv%29%29%5E2=abs%28abs%28u%29%29%5E2%2B2u%2Av%2Babs%28abs%28v%29%29%5E2
abs%28abs%28u-v%29%29%5E2=abs%28abs%28u%29%29%5E2-2u%2Av%2Babs%28abs%28v%29%29%5E2

By the equality
<=> 4u%2Av=0 <=>u%2Av=0
this means u and v are orthogonal