JEE MAIN - Mathematics (2003 - No. 25)
If $$\overrightarrow a \times \overrightarrow b = \overrightarrow b \times \overrightarrow c = \overrightarrow c \times \overrightarrow a $$ then $$\overrightarrow a + \overrightarrow b + \overrightarrow c = $$
$$abc$$
$$-1$$
$$0$$
$$2$$
Explanation
Let $$\overrightarrow a + \overrightarrow b + \overrightarrow c = \overrightarrow r .$$ Then
$$\overrightarrow a \times \left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right) = \overrightarrow a \times \overrightarrow r $$
$$ \Rightarrow 0 + \overrightarrow a \times \overrightarrow b + \overrightarrow a \times \overrightarrow c = \overrightarrow a \times \overrightarrow r $$
$$ \Rightarrow \overrightarrow a \times \overrightarrow b - \overrightarrow c \times \overrightarrow a = \overrightarrow a \times \overrightarrow r $$
$$ \Rightarrow \overrightarrow a \times \overrightarrow r = \overrightarrow 0 $$
Similarly $$\overrightarrow b \times \overrightarrow r = \overrightarrow 0 \,\,\,\& \,\,\,\overrightarrow c \times \overrightarrow r = \overrightarrow 0 $$
Above three conditions will be satisfied for non-zero vectors if and only if $$\overrightarrow r = \overrightarrow 0 $$
$$\overrightarrow a \times \left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right) = \overrightarrow a \times \overrightarrow r $$
$$ \Rightarrow 0 + \overrightarrow a \times \overrightarrow b + \overrightarrow a \times \overrightarrow c = \overrightarrow a \times \overrightarrow r $$
$$ \Rightarrow \overrightarrow a \times \overrightarrow b - \overrightarrow c \times \overrightarrow a = \overrightarrow a \times \overrightarrow r $$
$$ \Rightarrow \overrightarrow a \times \overrightarrow r = \overrightarrow 0 $$
Similarly $$\overrightarrow b \times \overrightarrow r = \overrightarrow 0 \,\,\,\& \,\,\,\overrightarrow c \times \overrightarrow r = \overrightarrow 0 $$
Above three conditions will be satisfied for non-zero vectors if and only if $$\overrightarrow r = \overrightarrow 0 $$
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