Question #214072

Suppose u, v \in V and ||u|| = ||v|| = 1 with < u,v > = 1: Prove that u = v.


1
Expert's answer
2021-07-08T06:47:53-0400

Any two or more vectors will be equal if they are collinear, codirected, and have the same magnitude.

Given


u=v=1||u||=||v||=1

u,v=1=>uvcos(u,v)=1\langle u, v\rangle=1=>||u||\cdot||v||\cdot\cos\angle(u, v)=1


=>cos(u,v)=1=>(u,v)=0=>\cos\angle(u, v)=1=>\angle(u, v)=0


=>=> uu and vv are collinear, codirected.

The unit vectors uu and vv have the same magnitude, are collinear, codirected.

Therefore the vectors uu and vv are equal.


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