lets-u-v-w-C-3-such-that-uvw-0-proof-that-or-give-a-counter-example-v-u-w-v-u-v-w-u-w-u-v-w-0-

Question Number 3410 by 123456 last updated on 13/Dec/15

lets (u, v, w) \in C^{3} such that u v w \neq 0, proof

that (or give a counter example)

\frac{v}{u} = \frac{w}{v} \land ∣ u ∣=∣ v ∣=∣ w ∣ \land u \neq w \Rightarrow u + v + w = 0

Commented by RasheedSindhi last updated on 13/Dec/15

W h a t i s m e a n t b y u, v, w \in C^{3} ?

Commented by 123456 last updated on 13/Dec/15

u \in C \land v \in C \land w \in C

Commented by RasheedSindhi last updated on 13/Dec/15

D o y o u m e a n

G i v e n : (u, v, w) \in C^{3} \land u v w \neq 0

T o p r o v e : \frac{v}{u} = \frac{w}{v} \land ∣ u ∣=∣ v ∣=∣ w ∣ \land u \neq w \Rightarrow u + v + w = 0 ?

Commented by 123456 last updated on 13/Dec/15

yes (or give a conter example)

Commented by Yozzi last updated on 13/Dec/15

v^{2} = u w . v =∣ v ∣ e^{i θ}, u =∣ u ∣ e^{i α}, w =∣ w ∣ e^{i β}

∣ v ∣^{2} e^{2 i θ} =∣ u ∣∣ w ∣ e^{i (α + β)} =∣ v ∣^{2} e^{i (α + β)}

e^{2 i θ} = e^{i (α + β)}

\Rightarrow c o s 2 θ = c o s (α + β)

s i n 2 θ = s i n (α + β)

u \neq w \Rightarrow α \neq β .

s i n 2 θ = s i n (α + β) :

2 c o s \frac{2 θ + α + β}{2} s i n \frac{2 θ - α - β}{2} = 0

\Rightarrow 2 θ - α - β = 2 n π \Rightarrow θ = \frac{α + β}{2} + n π n \in Z

\frac{2 θ + α + β}{2} = 2 r π \pm 0.5 π

2 θ + α + β = (4 r \pm 1) π

θ = \frac{(4 r \pm 1) π - α - β}{2} r \in Z

- π < θ ⩽ π .

\frac{v}{u} = \frac{w}{v} \Rightarrow e^{i (θ - α)} = e^{i (β - θ)}

\Rightarrow c o s (θ - α) = c o s (β - θ)

s i n (θ - α) = s i n (β - θ)

l e t θ = 0, α = π / 2, β = - π / 2 (α \neq β)

∴ v + w + u =∣ v ∣ + ∣ w ∣ (0 - i) + ∣ u ∣ (0 + i)

=∣ v ∣ + i (- ∣ w ∣ + ∣ u ∣)

=∣ v ∣\neq 0 i f \frac{v}{u} = \frac{w}{v} a n d ∣ v ∣=∣ w ∣=∣ u ∣ .

u v w =∣ v ∣^{3} \times (- i) i =∣ v ∣^{3} \neq 0

R \subset C s o v =∣ v ∣\in C, u =∣ u ∣ i \in C, w = - ∣ w ∣ i \in C

W e c a n s e t ∣ v ∣=∣ w ∣=∣ u ∣= r f o r r > 0, r \in R .

Commented by Yozzii last updated on 13/Dec/15

T e s t i n g n e w f e a t u r e

Answered by prakash jain last updated on 13/Dec/15

Counter Example from {Yozzi}^{'} s comment

a, - a i, a i