$$\overrightarrow a ,\,\overrightarrow b $$  and $$\overrightarrow c $$ are three vectors with magnitude $$\left| {\overrightarrow a } \right| = 4,\,\left| {\overrightarrow b } \right| = 4,\,\left| {\overrightarrow c } \right| = 2$$       and such that $$\overrightarrow a $$ is perpendicular to $$\left( {\overrightarrow b + \overrightarrow c } \right),\,\overrightarrow b $$   is perpendicular to $$\left( {\overrightarrow c + \overrightarrow a } \right)$$   and $$\overrightarrow c $$ is perpendicular to $$\left( {\overrightarrow a + \overrightarrow b } \right).$$   It follows that $$\left| {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right|$$    is equal to :

Solution :
Since, $$\overrightarrow a ,\,\overrightarrow b $$  and $$\overrightarrow c $$ are three vectors with magnitude $$\left| {\overrightarrow a } \right| = \left| {\overrightarrow b } \right| = 4{\text{ and }}\left| {\overrightarrow c } \right| = 2$$
As $$\overrightarrow a $$ is perpendicular to $$\left( {\overrightarrow b + \overrightarrow c } \right)$$
$$ \Rightarrow \overrightarrow a .\left( {\overrightarrow b + \overrightarrow c } \right) = 0{\text{ or }}\overrightarrow a .\overrightarrow b + \overrightarrow a .\overrightarrow c = 0......\left( {\text{i}} \right)$$
$$\overrightarrow b $$ is perpendicular to $$\left( {\overrightarrow c + \overrightarrow a } \right)$$
$$ \Rightarrow \overrightarrow b .\left( {\overrightarrow c + \overrightarrow a } \right) = 0{\text{ or }}\overrightarrow b .\overrightarrow c + \overrightarrow b .\overrightarrow a = 0......\left( {{\text{ii}}} \right)$$
$$\overrightarrow c $$ is perpendicular to $$\left( {\overrightarrow a + \overrightarrow b } \right)$$
$$ \Rightarrow \overrightarrow c .\left( {\overrightarrow a + \overrightarrow b } \right) = 0{\text{ or }}\overrightarrow c .\overrightarrow a + \overrightarrow c .\overrightarrow b = 0......\left( {{\text{iii}}} \right)$$
From equations $$\left( {\text{i}} \right),\,\left( {{\text{ii}}} \right)$$  and $$\left( {{\text{iii}}} \right),$$  we get
$$ \Rightarrow 2\left( {\overrightarrow a .\overrightarrow b + \overrightarrow b .\overrightarrow c + \overrightarrow c .\overrightarrow a } \right) = 0$$
Further we know that
$$\eqalign{ & {\left| {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right|^2} = {\left| {\overrightarrow a } \right|^2} + {\left| {\overrightarrow b } \right|^2} + {\left| {\overrightarrow c } \right|^2} + \overrightarrow {2a} .\overrightarrow b + \overrightarrow {2b} .\overrightarrow c + \overrightarrow {2c} .\overrightarrow a \cr & \Rightarrow {\left| {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right|^2} = {4^2} + {4^2} + {2^2} + 0 \cr & \Rightarrow {\left| {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right|^2} = 36 \cr & {\text{or }}\left| {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right| = 6 \cr} $$