Q78 / 125 · BITSAT 2013 · Mathematics

If $\vec{a}, \vec{b}, \vec{c}$ are three unit vectors such that $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0},$ where $\overrightarrow{0}$ is null vector, then $\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$ is :

  • A
    -3
  • B
    -2
  • Correct answer
    C
    $-\frac{3}{2}$
  • D
    0

Solution

We have $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$ $|\vec{a}+\vec{b}+\vec{c}|=0 \Rightarrow|\vec{a}+\vec{b}+\vec{c}|^{2}=0$ $\Rightarrow|\vec{a}|^{2}+|\vec{b}|^{2}+|\vec{c}|^{2}$ $+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})=0$ $\Rightarrow 1+1+1+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})=0$ $\Rightarrow \vec{a} \cdot \vec{b}+\bar{b} \cdot \vec{c}+\bar{c} \cdot \bar{a}=-\frac{3}{2}$

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

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