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)]
$=| a |^{2}+| b |^{2}+| c |^{2}$