Question

If $a,b$ and $c$ are three non-zero vectors such that each one of them being perpendicular to the sum of the other two vectors, then the value of $|a+b+c{|}^2$ is

• A. ${\left|a\right|}^2+{\left|b\right|}^2+{\left|c\right|}^2$
• B. $\left|a\right|+\left|b\right|+\left|c\right|$
• C. $2\left({\left|a\right|}^2+{\left|b\right|}^2+{\left|c\right|}^2\right)$
• D. $\frac{1}{2}\left({\left|a\right|}^2+{\left|b\right|}^2+{\left|c\right|}^2\right)$
• E. 0

Answer 1

Answer: (A)

According to the given condition, each vector is perpendicular to the sum of two vectors.
$\therefore a\cdot (b+c)=0,$
$b\cdot (a+c)=0$
and $c\cdot (a+b)=0$,
$\Rightarrow a\cdot b+a\cdot c=0,b\cdot a+b\cdot c=0$
and $c\cdot a+c\cdot b=0$
$\Rightarrow 2(a\cdot b+b\cdot c+c\cdot a)=0...(i)$
$Now,|a+b+c{|}^2=|a{|}^2+|b{|}^2+|c{|}^2+2(a\cdot b+b\cdot c+c\cdot a)$
$=|a{|}^2+|b{|}^2+|c{|}^2+2(0)[$ From Eq. (i)]
$=|a{|}^2+|b{|}^2+|c{|}^2$