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# If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ represent the vertices of a triangle,then show that $\frac{1}{2}\mid\overrightarrow{b}\times\overrightarrow{c}+\overrightarrow{c}\times\overrightarrow{a}+\overrightarrow{a}\times\overrightarrow{b}\mid$ gives the vector area of the triangle .Hence deduce the condition that the three points $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are collinear.Also find the unit vector normal to the plane of the triangle.

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• In a triangle$ABC$ $\overrightarrow a+\overrightarrow b+\overrightarrow c=0$
• Area of $\Delta=\large \frac{1}{2}$$|\overrightarrow {AB} \times \overrightarrow {AC}| Let \overrightarrow {OA}=\overrightarrow a,\overrightarrow {OB}=\overrightarrow{b} \;and \; \overrightarrow{OC}=\overrightarrow{c} \overrightarrow{AB}=\overrightarrow{b}-\overrightarrow{a} \overrightarrow{AC}=\overrightarrow{c}-\overrightarrow{a} =>\overrightarrow{AB} \times \overrightarrow{AC}=(\overrightarrow{b}-\overrightarrow{a}) \times (\overrightarrow{c}-\overrightarrow{a}) On expanding we get, =\overrightarrow{b}\times\overrightarrow{c}-\overrightarrow{b}\times\overrightarrow{a}-\overrightarrow{a}\times\overrightarrow{c}+\overrightarrow{a}\times\overrightarrow{a} But \overrightarrow{a}\times\overrightarrow{a}=0 and \overrightarrow{-b}\times\overrightarrow{a}=\overrightarrow{a}\times\overrightarrow{b} and \overrightarrow{-a}\times\overrightarrow{c}=\overrightarrow{c}\times\overrightarrow{a} Therefore \overrightarrow {AB} \times \overrightarrow {AC}=\overrightarrow b \times \overrightarrow c+\overrightarrow a \times \overrightarrow b+\overrightarrow c\times \overrightarrow a+\overrightarrow {0} \qquad\qquad\qquad\qquad=\overrightarrow b \times \overrightarrow c+\overrightarrow a \times \overrightarrow b+\overrightarrow c\times \overrightarrow a Step 2: Area of \Delta \; ABC=\large \frac{1}{2}$$|\overrightarrow {AB} \times \overrightarrow {AC}|$
$\qquad\qquad\qquad= \large\frac{1}{2}$$|\overrightarrow a\times \overrightarrow b+\overrightarrow b \times \overrightarrow c+\overrightarrow c \times \overrightarrow a| If the points are collinear then area of the triangle is 0 => \large\frac{1}{2}$$|\overrightarrow a\times \overrightarrow b+\overrightarrow b \times \overrightarrow c+\overrightarrow c \times \overrightarrow a|=0$
$|\overrightarrow a\times \overrightarrow b+\overrightarrow b \times \overrightarrow c+\overrightarrow c \times \overrightarrow a|=0$
$\overrightarrow a\times \overrightarrow b+\overrightarrow b \times \overrightarrow c+\overrightarrow c \times \overrightarrow a=0$ is the required condition of collinearity of three points $\overrightarrow a,\overrightarrow b$ and$\overrightarrow c$