Let $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ be three non-coplanar vectors and let $\mathbf{p}, \mathbf{q}$…

Let $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ be three non-coplanar vectors and let $\mathbf{p}, \mathbf{q}$ and $\mathbf{r}$ be the vectors defined by $\mathbf{p}=\frac{\mathbf{b} \times \mathbf{c}}{[\mathbf{a} \mathbf{b} \mathbf{c}]}, \mathbf{q}=\frac{\mathbf{c} \times \mathbf{a}}{[\mathbf{a} \mathbf{b} \mathbf{c}]}, \mathbf{r}=\frac{\mathbf{a} \times \mathbf{b}}{[\mathbf{a} \mathbf{b} \mathbf{c}]}$. Then, $(\mathbf{a}+\mathbf{b}) \cdot \mathbf{p}+(\mathbf{b}+\mathbf{c}) \cdot \mathbf{q}+(\mathbf{c}+\mathbf{a}) \cdot \mathbf{r}$ is equal to
  1. $0$
  2. $1$
  3. $2$
  4. $3$

Solution

$\begin{aligned} & =1+0=1 \\ & \text { Similarly, } \quad(\mathbf{b}+\mathbf{c}) \cdot \mathbf{q}=1 \\ & \text { and } \quad(\mathbf{c}+\mathbf{a}) \cdot \mathbf{r}=1 \\ & \end{aligned}$ $\begin{gathered}\therefore(\mathbf{a}+\mathbf{b}) \cdot \mathbf{p}+(\mathbf{b}+\mathbf{c}) \cdot \mathbf{q}+(\mathbf{c}+\mathbf{a}) \cdot \mathbf{r} \\ =1+1+1=3\end{gathered}$

Asked in: AP EAMCET 2012

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