Question

If $a = 3, b = 4, c = 5$ each one of $a ,b$ & $c$ is perpendicular to the sum of the remaining then $| a + b + c |$ is equal to

• A. $\frac{5}{\sqrt{2}}$
• B. $\frac{2}{\sqrt{5}}$
• C. $5 \sqrt{2}$
• D. $\sqrt{5}$

Answer 1

Answer: (C)

$5 \sqrt{2}$

Answer 2

We have, $| a |=3,| b |=4,| c |=5$
and $a \cdot b + b \cdot c + c \cdot a =0$
$\Rightarrow| a + b + c |^{2}=( a + b + c ) \cdot( a + b + c )$
$\Rightarrow| a + b + c |^{2}=| a |^{2}+| b |^{2}+| c |^{2}$
$+2( a \cdot b + b \cdot c + c \cdot a )$
$\Rightarrow| a + b + c |^{2}=(3)^{2}+(4)^{2}+(5)^{2}+2(0)$
$\Rightarrow| a + b + c |^{2}=9+16+25=50$
$\therefore \quad| a + b + c |=\sqrt{50}=5 \sqrt{2}$