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))


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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

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