Question

Vector u and v are orthogonal if and only if u⃗ .v⃗ =​

Answer 1

Answer:

If u,vu,v are orthogonal vectors, then:

∥u+v∥2=∥u∥2+∥v∥2‖u+v‖2=‖u‖2+‖v‖2

∥u−v∥2=∥u∥2+∥−v∥2=∥u∥2+∥v∥2‖u−v‖2=‖u‖2+‖−v‖2=‖u‖2+‖v‖2

now ∥u+v∥2=∥u−v∥2‖u+v‖2=‖u−v‖2, but the norm is ever positive therefore: ∥u+v∥=∥u−v∥‖u+v‖=‖u−v‖.

=> Now, if ∥u+v∥=∥u−v∥‖u+v‖=‖u−v‖ we have:

∥u+v∥2=∥u∥2+2u⋅v+∥v∥2‖u+v‖2=‖u‖2+2u⋅v+‖v‖2

∥u−v∥2=∥u∥2−2u⋅v+∥v∥2

Step-by-step explanation:

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