Question

The magnitude of a vector which is orthogonal to the vector \[\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\] and is coplanar with the vectors \[\hat{\mathrm{i}}+\hat{\mathrm{j}}+2\hat{\mathrm{k}}\] and \[\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\hat{\mathrm{k}}\] is

Options

• √2

• 4√2

• 4

• 2√3

Answer 1

4√2

Explanation:

Let, \[\overline{\mathrm{a}}\] = \[\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\], \[\overline{\mathrm{b}}\] = \[\hat{\mathrm{i}}+\hat{\mathrm{j}}+2\hat{\mathrm{k}}\] and \[\overline{\mathrm{c}}\] =\[\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\hat{\mathrm{k}}\] 

By definition, a vector orthogonal to \[\overline{\mathrm{a}}\] and coplanar to \[\overline{\mathrm{b}}\] and \[\overline{\mathrm{c}}\] is given by \[\overline{\mathrm{a}}\times\left(\overline{\mathrm{b}}\times\overline{\mathrm{c}}\right)\]

\[= \begin{pmatrix} \overline{\mathrm{a}}.\overline{\mathrm{c}} \end{pmatrix}\overline{\mathrm{b}}- \begin{pmatrix} \overline{\mathrm{a}}.\overline{\mathrm{b}} \end{pmatrix}\overline{\mathrm{c}}\]

\[=4\left(\hat{\mathrm{i}}+\hat{\mathrm{j}}+2\hat{\mathrm{k}}\right)-4\left(\hat{\mathrm{j}}+2\hat{\mathrm{j}}+\hat{\mathrm{k}}\right)\]

\[=4\left(-\hat{\mathbf{j}}+\hat{\mathbf{k}}\right)\]

\[\therefore\quad\mathrm{Magnitude}=\sqrt{4^2\left(1^2+1^2\right)}=4\sqrt{2}\]