If $\hat{\mathbf{a}}, \hat{\mathbf{b}}$ and $\hat{\mathbf{c}}$ are unit vectors such that…

If $\hat{\mathbf{a}}, \hat{\mathbf{b}}$ and $\hat{\mathbf{c}}$ are unit vectors such that $\hat{\mathbf{a}}+\hat{\mathbf{b}}+\hat{\mathbf{c}}=\mathbf{0}$, then the $\hat{\mathbf{a}} \cdot \hat{\mathbf{b}}+\hat{\mathbf{b}} \cdot \hat{\mathbf{c}}+\hat{\mathbf{c}} \cdot \hat{\mathbf{a}}$ is equal to
  1. $\frac{3}{2}$
  2. $\frac{-3}{2}$
  3. $\frac{1}{2}$
  4. $\frac{-1}{2}$

Solution

Given that $\hat{\mathrm{a}}, \hat{\mathrm{b}}, \hat{\mathrm{c}}$ are unit vectors and $\hat{\mathrm{a}}+\hat{\mathrm{b}}+\hat{\mathrm{c}}=0$ $\therefore \quad|\hat{a}|=|\hat{b}|=|\hat{c}|=1$ Here, $\hat{a}+\hat{b}+\hat{c}=\hat{0}$ Then, $(\hat{a}+\hat{b}+\hat{c})^2=(\hat{0})^2$ $\begin{aligned} & \Rightarrow \quad(\hat{a}+\hat{b}+\hat{c}) \cdot(\hat{a}+\hat{b}+\hat{c})=0 \\ & \Rightarrow \quad \hat{a} \cdot \hat{a}+\hat{a} \cdot \hat{b}+\hat{a} \cdot \hat{c}+\hat{b} \cdot \hat{a}+\hat{b} \cdot \hat{b} \\ & \hat{b} \cdot \hat{c}+\hat{c} \cdot \hat{a}+\hat{c} \cdot \hat{b}+\hat{c} \cdot \hat{c}=0 \\ & \Rightarrow \quad|\hat{a}|^2+\hat{a} \cdot \hat{b}+\hat{c} \cdot \hat{a}+\hat{a} \cdot \hat{b}+|\hat{b}|^2+\hat{b} \cdot \hat{c} \\ & +\hat{c} \cdot \hat{a}+\hat{b} \cdot \hat{c}+|\hat{c}|^2=0 \\ & \Rightarrow \quad(1)^2+2 \hat{a} \cdot \hat{b}+2 \hat{b} \cdot \hat{c}+2 \hat{c} \cdot \hat{a}+(1)^2+(1)^2=0 \\ & \Rightarrow \quad 2[\hat{a} \cdot \hat{b}+\hat{b} \cdot \hat{c}+\hat{c} \cdot \hat{a}]+3=0 \\ & 2[\hat{a} \cdot \hat{b}+\hat{b} \cdot \hat{c}+\hat{c} \cdot \hat{a}]=-3 \\ & \hat{a} \cdot \hat{b}+\hat{b} \cdot \hat{c}+\hat{c} \cdot \hat{a}=-\frac{3}{2} \end{aligned}$

Asked in: AP EAMCET 2015

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