Question

If \[\overline{\mathrm{a}}=4\hat{\mathrm{i}}+3\hat{\mathrm{j}}+\hat{\mathrm{k}},\overline{\mathrm{b}}=\hat{\mathrm{i}}-2\hat{\mathrm{j}}+2\hat{\mathrm{k}}\] then \[\mathbf{\overline{a}}\times\left(\mathbf{\overline{a}}\times\left(\mathbf{\overline{a}}\times\mathbf{\overline{b}}\right)\right)=\]

पर्याय

• 676\[\overline{a}\]

• 676\[\overline{b}\]

• 625\[\overline{a}\]

• 625\[\overline{b}\]

Answer 1

676\[\overline{b}\]

Explanation:

\[\overset{-}{\operatorname*{\mathbf{a}}}\cdot\overset{-}{\operatorname*{\mathbf{b}}}=\left(4\hat{\mathbf{i}}+3\hat{\mathbf{j}}+\hat{\mathbf{k}}\right)\cdot\left(\hat{\mathbf{i}}-2\hat{\mathbf{j}}+2\hat{\mathbf{k}}\right)=0\]

\[\begin{array} {cc}\therefore & \bar{\mathrm{a}}\perp\bar{\mathrm{b}} \end{array}\]

\[\overset{-}{\operatorname*{\mathrm{a}}}\times\left(\overset{-}{\operatorname*{\mathrm{a}}}\times\overset{-}{\operatorname*{\mathrm{b}}}\right)=\left(\overset{-}{\operatorname*{\mathrm{a}}}\cdot\overset{-}{\operatorname*{\mathrm{b}}}\right)\overset{-}{\operatorname*{\mathrm{a}}}-\left(\overset{-}{\operatorname*{\mathrm{a}}}\cdot\overset{-}{\operatorname*{\mathrm{a}}}\right)\overline{\mathrm{b}}\]

\[=-\left|\overline{\mathrm{a}}\right|^{2}\overline{\mathrm{b}}\]

\[\overset{-}{\operatorname*{\operatorname*{a}}}\times\left(\overset{-}{\operatorname*{\operatorname*{a}}}\times\left(-\left|\overset{-}{\operatorname*{\operatorname*{a}}}\right|^2\overset{-}{\operatorname*{\operatorname*{b}}}\right)\right)=-\left|\overset{-}{\operatorname*{\operatorname*{a}}}\right|^2\left(\overset{-}{\operatorname*{\operatorname*{a}}}\times\left(\overset{-}{\operatorname*{\operatorname*{a}}}\times\overset{-}{\operatorname*{\operatorname*{b}}}\right)\right)\]

\[=-|\overset{-}{\operatorname*{\operatorname*{a}}}|^2\left(-|\overset{-}{\operatorname*{\operatorname*{a}}}|^2\overset{-}{\operatorname*{\operatorname*{b}}}\right)\]

\[=\left|\overline{\mathrm{a}}\right|^4\overline{\mathrm{b}}\]

\[\begin{vmatrix} - \\ a \end{vmatrix}=\sqrt{16+9+1}=\sqrt{26}\]

\[\therefore\quad\left|\overline{\mathrm{a}}\right|^4=676\]