If $\vec a=\hat i+2\hat j+3\hat k,\vec b=2\hat i+3\hat j+2\hat k,\vec c$ is any vector not collinear with $\vec b$, then magnitude of $\frac{\vec a.(\vec b×\vec c)}{|\vec b×\vec c|^2}(\vec b×\vec c)+\left\{\frac{\vec a.\vec b}{|\vec b|^2}\right\}\vec b+\left\{\frac{\vec a.\vec c}{|\vec c|^2}\right\}\vec c$ is equal to

none of these

Clearly, $\vec b,\vec c$ and $\vec b×\vec c$ are non-coplanar vectors.

$∴\vec a=\left\{\frac{\vec a.\vec b}{|\vec b|^2}\right\}\vec b+\left\{\frac{\vec a.\vec c}{|\vec c|^2}\right\}\vec c+\frac{\vec a.(\vec b×\vec c)}{|\vec b×\vec c|^2}(\vec b×\vec c)$

Hence, the magnitude of the given vector =$|\vec a|=\sqrt{14}$.