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Home  >>  TN XII Math  >>  Vector Algebra
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if $\ \overrightarrow{a}= \overrightarrow{2i}+ \overrightarrow{3j}- \overrightarrow{k} , \overrightarrow{b}= -\overrightarrow{2i}+ \overrightarrow{5k}, \overrightarrow{c}= \overrightarrow{j}- \overrightarrow{3k}. $ Verify that $ \overrightarrow{a}\times ( \overrightarrow{b}\times \overrightarrow{c})= (\overrightarrow{a}. \overrightarrow{c}) \overrightarrow{b}- (\overrightarrow{a}. \overrightarrow{b})c$

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  • If $ \overrightarrow a = a_1\overrightarrow i + a_2 \overrightarrow j+a_3 \overrightarrow k,\: \: \overrightarrow b = b_1 \overrightarrow i+b_2\overrightarrow j + b_3 \overrightarrow k$ then $ \overrightarrow a.\overrightarrow b = a_1b_1+a_2b_2+a_3b_3$
  • If $ \overrightarrow a = a_1\overrightarrow i+a_2\overrightarrow j+a_3\overrightarrow k, \: \overrightarrow b = b_1\overrightarrow i+b_2\overrightarrow j+b_3\overrightarrow k$ then $ \overrightarrow a$ x $ \overrightarrow b = \begin{vmatrix} \overrightarrow i & \overrightarrow j & \overrightarrow k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$
LHS To find $ \overrightarrow a \times ( \overrightarrow b \times \overrightarrow c)$ $ \overrightarrow b \times \overrightarrow c = \begin{vmatrix} \overrightarrow i & \overrightarrow j & \overrightarrow k \\ -2 & 0 & 5 \\ 0 & 1 & -3 \end{vmatrix} = -5\overrightarrow i-6\overrightarrow j-2\overrightarrow k$
$\overrightarrow a \times (\overrightarrow b \times \overrightarrow c) = \begin{vmatrix} \overrightarrow i & \overrightarrow j & \overrightarrow k \\ 2 & 3 & -1 \\ -5 & -6 & -2 \end{vmatrix} = (-6-6)\overrightarrow i-(-4-5)\overrightarrow j+(-12+15)\overrightarrow k$
$ = -12\overrightarrow i+9\overrightarrow j+3\overrightarrow k$
RHS To find $(\overrightarrow a.\overrightarrow c)\overrightarrow b-(\overrightarrow a.\overrightarrow b)\overrightarrow c$
$ \overrightarrow a.\overrightarrow b = (2)(-2)+3(0)+(-1)(5)=-4-5=-9$
$ \overrightarrow a.\overrightarrow c = (2)(0)+3(1)+(-1)(-3)=4$
$ (\overrightarrow a.\overrightarrow c)\overrightarrow b-(\overrightarrow a-\overrightarrow b)\overrightarrow c=4\overrightarrow b+9\overrightarrow c$
$ = -12\overrightarrow i+30\overrightarrow k+9\overrightarrow j-27\overrightarrow k=-12\overrightarrow i+9\overrightarrow j+3\overrightarrow k$
LHS = RHS
answered Jun 8, 2013 by thanvigandhi_1
 

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