prove-or-disprove-with-counter-example-that-a-For-all-two-dimensional-vectors-a-b-c-a-b-a-c-b-c-b-For-all-positive-real-numbers-a-b-a-b-2-ab-

Question Number 83965 by Rio Michael last updated on 08/Mar/20

prove or disprove (with counter - example) that

a) For all two dimensional vectors a, b, c,

a . b = a . c \Rightarrow b = c .

b) For all positive real numbers a, b .

\frac{a + b}{2} ⩾ \sqrt{a b}

Commented by mr W last updated on 08/Mar/20

(a)

s t a t e m e n t i s w r o n g!

a \cdot b =∣ a ∣∣ b ∣ \cos β

a \cdot c =∣ a ∣∣ c ∣ \cos γ

a \cdot b = a \cdot c \Rightarrow∣ b ∣ \cos β =∣ c ∣ \cos γ ⇏ b = c

e x a m p l e

a = (1, 0)

b = (1, 1)

c = (1, 2)

a \cdot b = a \cdot c = 1

b u t b \neq c

Commented by mr W last updated on 08/Mar/20

(b)

f o r a, b > 0 :

{(\sqrt{a} - \sqrt{b})}^{2} ⩾ 0

\Rightarrow a - 2 \sqrt{a b} + b ⩾ 0

\Rightarrow \frac{a + b}{2} ⩾ \sqrt{a b}

Commented by Rio Michael last updated on 08/Mar/20

p e r f e c t s i r, a n d t h a n k s f o r t h e c o r r e c t i o n