If $\overrightarrow a = 2\widehat i + \widehat j + 2\widehat k$, then the value of

$${\left| {\widehat i \times \left( {\overrightarrow a \times \widehat i} \right)} \right|^2} + {\left| {\widehat j \times \left( {\overrightarrow a \times \widehat j} \right)} \right|^2} + {\left| {\widehat k \times \left( {\overrightarrow a \times \widehat k} \right)} \right|^2}$$ is equal to____

Let $\overrightarrow a = x\widehat i + y\widehat j + z\widehat k$<br><br>Now $$\widehat i \times \left( {\overrightarrow a \times \widehat i} \right) = \left( {\widehat i.\widehat i} \right)\overrightarrow a - \left( {\widehat i.\overrightarrow a } \right)\widehat i$$<br><br>= $y\widehat j + z\widehat k$<br><br>Similarly $$\widehat j \times \left( {\overrightarrow a \times \widehat j} \right) = x\widehat i + z\widehat k$$<br><br>$$\widehat k \times \left( {\overrightarrow a \times \widehat k} \right) = x\widehat i + y\widehat j$$<br><br>Now $${\left| {y\widehat j + z\widehat k} \right|^2} + {\left| {x\widehat i + z\widehat k} \right|^2} + {\left| {x\widehat i + y\widehat j} \right|^2}$$<br><br>= $2({x^2} + {y^2} + {z^2})$ <br><br>Given $\overrightarrow a = 2\widehat i + \widehat j + 2\widehat k$ <br><br>$\therefore$ x = 2, y = 1, z = 2 <br><br>= 2(4 + 1 + 4) = 18