Question

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

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

Answer 1

Answer: (C)

We have, $âˆ\pounds aâˆ\pounds =3,âˆ\pounds bâˆ\pounds =4,âˆ\pounds câˆ\pounds =5$
and $aâ‹\dots b+bâ‹\dots c+câ‹\dots a=0$
$⇒âˆ\pounds a+b+c{âˆ\pounds }^2=(a+b+c)â‹\dots (a+b+c)$
$⇒âˆ\pounds a+b+c{âˆ\pounds }^2=âˆ\pounds a{âˆ\pounds }^2+âˆ\pounds b{âˆ\pounds }^2+âˆ\pounds c{âˆ\pounds }^2$
$+2(aâ‹\dots b+bâ‹\dots c+câ‹\dots a)$
$⇒âˆ\pounds a+b+c{âˆ\pounds }^2=(3{)}^2+(4{)}^2+(5{)}^2+2(0)$
$⇒âˆ\pounds a+b+c{âˆ\pounds }^2=9+16+25=50$
$∴âˆ\pounds a+b+câˆ\pounds =\sqrt{50}=5\sqrt{2}$