Trying to prove: Show that there are no vectors u and v such that ∥u∥=1, ∥v∥=2, and u*v=3.

Melina Barber 2022-09-19 Answered
Trying to prove:
Show that there are no vectors u and v such that u = 1, v = 2, and u v = 3
Don't know where to go from here:
u = 2 u 2 = 4 u u = 1
v = 1 v 2 = 1 v v = 4
Not sure if this is the right direction to take, but we have:
u v = v v u u = 3 = ( v + u ) ( v u )
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Answers (2)

Santiago Collier
Answered 2022-09-20 Author has 8 answers
u v = u v cos θ
So
3 = ( 1 ) ( 2 ) cos θ
which implies
cos θ = 3 2
which is not possible.

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easternerjx
Answered 2022-09-21 Author has 3 answers
Another way to realize the answer:
u + v 2 = u + v , u + v = u , u + 2 u , v + v , v = u 2 + 2 u , v + v 2 = 1 2 + 2 × 3 + 2 2 = 1 + 6 + 4 > ( 1 + 2 ) 2 = ( u + v ) 2
which violates the triangle inequality.

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