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