$$ \text { Let } \vec{a}=2 \hat{i}-\hat{j}+5 \hat{k} \text { and } \vec{b}=\alpha \hat{i}+\beta \hat{j}+2 \hat{k} \text {. If }((\vec{a} \times \vec{b}) \times \hat{i}) \cdot \hat{k}=\frac{23}{2} \text {, then }|\vec{b} \times 2 \hat{j}| $$ is equal to :

<p>Given, $\overrightarrow a = 2\widehat i - \widehat j + 5\widehat k$ and $\overrightarrow b = \alpha \widehat i + \beta \widehat j + 2\widehat k$</p> <p>Also, $$\left( {\left( {\overrightarrow a \times \overrightarrow b } \right) \times i} \right)\,.\,\widehat k = {{23} \over 2}$$</p> <p>$$ \Rightarrow \left( {\left( {\overrightarrow a \,.\,\widehat i} \right)\overrightarrow b - \left( {\overrightarrow b \,.\,\widehat i} \right)\,.\,\overline a } \right)\,.\,\widehat k = {{23} \over 2}$$</p> <p>$$ \Rightarrow \left( {2\,.\,\overrightarrow b - \alpha \,.\,\overrightarrow a } \right)\,.\,\widehat k = {{23} \over 2}$$</p> <p>$$ \Rightarrow 2\,.\,2 - 5\alpha = {{23} \over 2} \Rightarrow \alpha = {{ - 3} \over 2}$$</p> <p>Now, $$\left| {\overrightarrow b \times 2j} \right| = \left| {\left( {\alpha \widehat i + \beta \widehat j + 2\widehat k} \right) \times 2\widehat j} \right|$$</p> <p>$= \left| {2\alpha \widehat k + 0 - 4\widehat i} \right|$</p> <p>$= \sqrt {4{\alpha ^2} + 16}$</p> <p>$= \sqrt {4{{\left( {{{ - 3} \over 2}} \right)}^2} + 16} = 5$</p>