JEE MAIN - Mathematics (2002 - No. 29)
If the vectors $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ from the sides $B C, C A$ and $A B$ respectively of a triangle $A B C$, then :
$\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=\overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{c}} \cdot \overrightarrow{\mathbf{b}}=0$
$\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}=\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}$
$\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=\overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{c}} \cdot \overrightarrow{\mathbf{a}}=0$
$\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}=\overrightarrow{\mathbf{0}}$
Explanation
If $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ are the sides of $\mathbf{a}$ triangle, then
$\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{0}}$
Since,
$$ \begin{aligned} \vec{a}+\vec{b}+\vec{c} & =\overrightarrow{0} \\\\ \vec{a}+\vec{b} & =-\vec{c} \end{aligned} $$
$$ \begin{array}{ll} \Rightarrow & (\vec{a}+\vec{b}) \times \vec{c}=-\vec{c} \times \vec{c} \\\\ \Rightarrow & \vec{a} \times \vec{c}+\vec{b} \times \vec{c}=\overrightarrow{0} \\\\ \Rightarrow & \vec{b} \times \vec{c}=\vec{c} \times \vec{a} \\\\ \text { Similarly, } & \vec{a} \times \vec{b}=\vec{b} \times \vec{c} \\\\ \text { Hence, } & \vec{a} \times \vec{b}=\vec{b} \times \vec{c}=\vec{c} \times \vec{a} \end{array} $$
$\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{0}}$
Since,
$$ \begin{aligned} \vec{a}+\vec{b}+\vec{c} & =\overrightarrow{0} \\\\ \vec{a}+\vec{b} & =-\vec{c} \end{aligned} $$
$$ \begin{array}{ll} \Rightarrow & (\vec{a}+\vec{b}) \times \vec{c}=-\vec{c} \times \vec{c} \\\\ \Rightarrow & \vec{a} \times \vec{c}+\vec{b} \times \vec{c}=\overrightarrow{0} \\\\ \Rightarrow & \vec{b} \times \vec{c}=\vec{c} \times \vec{a} \\\\ \text { Similarly, } & \vec{a} \times \vec{b}=\vec{b} \times \vec{c} \\\\ \text { Hence, } & \vec{a} \times \vec{b}=\vec{b} \times \vec{c}=\vec{c} \times \vec{a} \end{array} $$
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