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Home  >>  CBSE XII  >>  Math  >>  Vector Algebra
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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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Toolbox:
  • 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$
answered May 30, 2013 by meena.p
 

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