Q99 / 180 · KCET 2013 · Mathematics

If $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are unit vectors such that $\mathbf{a}+\mathbf{b}+\mathbf{c}=0$, then $\mathbf{a} \cdot \mathbf{b}+\mathbf{b} \cdot \mathbf{c}+\mathbf{c} \cdot \mathbf{a}$ is equal to

  • A
    $\frac{3}{2}$
  • Correct answer
    B
    $-\frac{3}{2}$
  • C
    $\frac{2}{3}$
  • D
    $\frac{1}{2}$

Solution

Given that, $a, b$ and $c$ are unit vectors. i.e., $\quad|a|=|b|=|c|=1...(i)$ Now, we have $\begin{gathered} (a+b+c)=0 \\ \Rightarrow \quad|a+b+c|^{2}=|0|^{2} \\ \Rightarrow \quad|a|^{2}+|b|^{2}+|c|^{2}+2(a \cdot b+b \cdot c+c \cdot a)=0 \\ \Rightarrow \quad 1+1+1+2(a \cdot b+b \cdot c+c \cdot a)=0 [from Eq. (i)] \\ \Rightarrow \quad(a \cdot b+b \cdot c+c \cdot a)=-\frac{3}{2} \end{gathered}$

Hence, the correct answer is option: 'B'.

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