Mathematics - Vector Question with Solution | TestHub

From the given information, it is clear that $$\begin{array}{c}\overrightarrow{\mathbf{a}}=\frac{\hat{\mathbf{i}}-2\hat{\mathbf{j}}}{\sqrt{5}}\\ \Rightarrow |\overrightarrow{\mathbf{a}}|=1,|\overrightarrow{\mathbf{b}}|=1,\overrightarrow{\mathbf{a}}\cdot \overrightarrow{\mathbf{b}}=0\\ \text{ Now, }(2\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}})\cdot [(\overrightarrow{\mathbf{a}}\times \overrightarrow{\mathbf{b}})\times (\overrightarrow{\mathbf{a}}-2\overrightarrow{\mathbf{b}})]\end{array}$$ $$\begin{array}{rl}& =(2\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}})\cdot [a^2\overrightarrow{\mathbf{b}}-(\overrightarrow{\mathbf{a}}\cdot \overrightarrow{\mathbf{b}})\cdot \overrightarrow{\mathbf{a}}+2b^2\cdot \overrightarrow{\mathbf{a}}\\ & =[2\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}]\cdot [\overrightarrow{\mathbf{b}}+2\overrightarrow{\mathbf{a}}]=4{\overrightarrow{\mathbf{a}}}^2+{\overrightarrow{\mathbf{b}}}^2\\ & =4\cdot 1+1=5\text{ [as }a\cdot b=0]\end{array}$$