Let $\overrightarrow a = 2\widehat i - \widehat j + 2\widehat k$ and $\overrightarrow b = \widehat i + 2\widehat j - \widehat k$. Let a vector $\overrightarrow v$ be in the plane containing $\overrightarrow a$ and $\overrightarrow b$. If $\overrightarrow v$ is perpendicular to the vector $3\widehat i + 2\widehat j - \widehat k$ and its projection on $\overrightarrow a$ is 19 units, then ${\left| {2\overrightarrow v } \right|^2}$ is equal to _____________.

$\overrightarrow a = 2\widehat i - \widehat j + 2\widehat k$<br><br>$\overrightarrow b = \widehat i + 2\widehat j - \widehat k$<br><br>$\overrightarrow c = 3\widehat i + 2\widehat j - \widehat k$<br><br>$\overrightarrow v = x\overrightarrow a + y\overrightarrow b$<br><br>$\overrightarrow v \left( {3\widehat i + 2\widehat j - \widehat k} \right) = 0$<br><br>$\overrightarrow v .\widehat a = 19$<br><br>$$\overrightarrow v = \lambda \overrightarrow c \times \left( {\overrightarrow a \times \overrightarrow b } \right)$$<br><br>$$\overrightarrow v = \lambda \left[ {\left( {\overrightarrow c .\overrightarrow b } \right)\overrightarrow a - \left( {\overrightarrow c .\overrightarrow a } \right)\overrightarrow b } \right]$$<br><br>$$ = \lambda [(3 + 4 + 1)\left( {2\widehat i - \widehat j + 2\widehat k} \right) - \left( {{{6 - 2 - 2} \over 2}} \right)\left( {\widehat i + 2\widehat j + \widehat k} \right)$$<br><br>$$ = \lambda [16\widehat i - 8\widehat j + 16\widehat k - 2\widehat i - 4\widehat j + 2\widehat k]$$<br><br>$$\overrightarrow v = \lambda \left[ {14\widehat i - 12\widehat j + 18\widehat k} \right]$$<br><br>$$\lambda [14\widehat i - 12\widehat j + 18\widehat k].{{\left( {2\widehat i - \widehat j + 2\widehat k} \right)} \over {\sqrt {4 + 1 + 4} }} = 19$$<br><br>$\lambda {{[28 + 12 + 36]} \over 3} = 19$<br><br>$\lambda \left( {{{76} \over 3}} \right) = 19$<br><br>$4\lambda = 3 \Rightarrow \lambda = {3 \over 4}$<br><br>$$|2{v^2}| = {\left| {2 \times {3 \over 4}(14\widehat i - 12\widehat j + 18\widehat k)} \right|^2}$$<br><br>${9 \over 4} \times 4{\left( {7\widehat i - 6\widehat j + 9\widehat k} \right)^2}$<br><br>$= 9(49 + 36 + 81)$<br><br>$= 9(166)$<br><br>$= 1494$