Mathematics - Vector Question with Solution | TestHub

The volume of the parallelopiped with coterminus edges as $\hat{\mathbf{a}},\hat{\mathbf{b}},\hat{\mathbf{c}}$ is given by $[\hat{\mathbf{a}}\hat{\mathbf{b}}\hat{\mathbf{c}}]=\hat{\mathbf{a}}\cdot (\hat{\mathbf{b}}\times \hat{\mathbf{c}})$
$$\begin{array}{rl}& \text{ Now, }[\hat{\mathbf{a}}\hat{\mathbf{b}}\hat{\mathbf{c}}{]}^2=\left|\begin{array}{ccc}\hat{\mathbf{a}}\cdot \hat{\mathbf{a}}& \hat{\mathbf{a}}\cdot \hat{\mathbf{b}}& \hat{\mathbf{a}}\cdot \hat{\mathbf{c}}\\ \hat{\mathbf{b}}\cdot \hat{\mathbf{a}}& \hat{\mathbf{b}}\cdot \hat{\mathbf{b}}& \hat{\mathbf{b}}\cdot \hat{\mathbf{c}}\\ \hat{\mathbf{c}}\cdot \hat{\mathbf{a}}& \hat{\mathbf{c}}\cdot \hat{\mathbf{b}}& \hat{\mathbf{c}}\cdot \hat{\mathbf{c}}\end{array}\right|=\left|\begin{array}{ccc}1& 1/2& 1/2\\ 1/2& 1& 1/2\\ 1/2& 1/2& 1\end{array}\right|=\frac{1}{2}\\ & \Rightarrow [\hat{\mathbf{a}}\hat{\mathbf{b}}\hat{\mathbf{c}}{]}^2=\frac{1}{2}\\ & \end{array}$$ Thus, the required volume of the parallelopiped $=\frac{1}{\sqrt{2}}\mathrm{cu}$ units