Question

Three vectors $a , b$ and $c$ are such that $a + b + c =0$. If $| a |=1,| b |=4$ and $| c |=2$ and $\mu= a \cdot b + b \cdot c + c \cdot a$ then the value of $\mu$ is equal to

• A. 21
• B. $\frac{-21}{2}$
• C. $\frac{-11}{2}$
• D. $-11$

Answer 1

Answer: (B)

Since $a+b+c=0$, we have
$a \cdot(a+b+c)=0$
or $ a \cdot a+a \cdot b+a \cdot c=0$
Therefore, $ a \cdot b+a \cdot c=-|a|^2=-1 ...$(i)
Again $ b \cdot(a+b+c)=0$
or $ a \cdot b+b \cdot c=-|b|^2=-16 ...$(ii)
Similarly, $ a \cdot c+b \cdot c= -4...$(iii)
Adding (i), (ii) and (iii), we have
$2(a \cdot b+b \cdot c+a \cdot c) =-21 $
$2 \mu =-21 $
i.e., $\mu =\frac{-21}{2}$