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Home  >>  TN XII Math  >>  Vector Algebra
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Find$ (\overrightarrow{a}\times\overrightarrow{b}) . (\overrightarrow{c}\times\overrightarrow{d}) $if $\overrightarrow{a}=\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}, \overrightarrow{b}=\overrightarrow{2i}+\overrightarrow{k}, \overrightarrow{c}=\overrightarrow{2i}+\overrightarrow{j}+\overrightarrow{k}, \overrightarrow{d}=\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{2k}$

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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$
  • Scalar product of four vectors $ ( \overrightarrow a \times \overrightarrow b).(\overrightarrow c \times \overrightarrow a) = \begin{vmatrix} \overrightarrow a.\overrightarrow c & \overrightarrow a.\overrightarrow d \\ \overrightarrow b.\overrightarrow c & \overrightarrow b.\overrightarrow d \end{vmatrix}$
$ ( \overrightarrow a \times \overrightarrow b).(\overrightarrow c \times \overrightarrow d) = \begin{vmatrix} \overrightarrow a.\overrightarrow c & \overrightarrow a.\overrightarrow d \\ \overrightarrow b.\overrightarrow c & \overrightarrow b.\overrightarrow d \end{vmatrix} = (\overrightarrow a.\overrightarrow c)(\overrightarrow b.\overrightarrow d)-(\overrightarrow b.\overrightarrow c)(\overrightarrow a.\overrightarrow d)$
$\overrightarrow a.\overrightarrow c = (1)(2)+(1)(1)+(1)(1)=4$
$\overrightarrow a.\overrightarrow d = (1)(1)+(1)(1)+(1)(2)=4$
$\overrightarrow b.\overrightarrow c = (2)(2)+0+(1)(1)=5$
$\overrightarrow b.\overrightarrow d = (2)(1)+0+(1)(2)=4$
$ (\overrightarrow a \times \overrightarrow b).(\overrightarrow c \times \overrightarrow d) = 16-20=-4$
answered Jun 10, 2013 by thanvigandhi_1
 

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