Let a → = 2 i ^ - j ^ + 5 k ^ and b → = α i ^ + β j ^ + 2 k ^ . If a → × b…

Given, $\overrightarrow{a}=2\hat{i}-\hat{j}+5\hat{k},\overrightarrow{b}=\alpha \hat{i}+\beta \hat{j}+2\hat{k}$

Also given $\left(\left(\overrightarrow{a}\times \overrightarrow{b}\right)\times \hat{i}\right)·\hat{k}=\frac{23}{2}$, then $\left|\overrightarrow{b}\times 2\hat{j}\right|$ is

Now using triple cross product we have, $\left(\left(\overrightarrow{a}·\hat{i}\right)\overrightarrow{b}-\left(\overrightarrow{b}·\hat{i}\right)\overrightarrow{a}\right)·\hat{k}=\frac{23}{2}$

$\Rightarrow \left(2.\overrightarrow{b}-\alpha ·\overrightarrow{a}\right).\hat{k}=\frac{23}{2}$

$\Rightarrow 2\times 2-\alpha \times 5=\frac{23}{2}$

$\Rightarrow 5\alpha =4-\frac{23}{2}$

$\Rightarrow \alpha =\frac{-3}{2}$

$ \vec{b} \times 2\vec{j} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ \alpha & \beta & 2 \\ 0 & 2 & 0 \end{vmatrix} = -4\vec{i} + 2\alpha\vec{k} $ $ \therefore |\vec{b} \times 2\vec{j}| = \sqrt{16 + 4\alpha^2} = \sqrt{16 + 4 \times \frac{9}{4}} = 5 $