Let the vectors $\overrightarrow a$, $\overrightarrow b$, $\overrightarrow c$ be such that
$\left| {\overrightarrow a } \right| = 2$, $\left| {\overrightarrow b } \right| = 4$ and $\left| {\overrightarrow c } \right| = 4$. If the projection of
$\overrightarrow b$ on $\overrightarrow a$ is equal to the projection of $\overrightarrow c$ on $\overrightarrow a$
and $\overrightarrow b$ is perpendicular to $\overrightarrow c$, then the value of
$\left| {\overrightarrow a + \vec b - \overrightarrow c } \right|$ is ___________.

Projection of $\overrightarrow b$ on $\overrightarrow a$ = Projection of $\overrightarrow c$ on $\overrightarrow a$ <br><br>$\Rightarrow$ $${{\overrightarrow b .\overrightarrow a } \over {\left| {\overrightarrow a } \right|}} = {{\overrightarrow c .\overrightarrow a } \over {\left| {\overrightarrow a } \right|}}$$ <br><br>$\Rightarrow$ $\overrightarrow b .\overrightarrow a = \overrightarrow c .\overrightarrow a$ <br><br>$\because$ $\overrightarrow b$ is perpendicular to $\overrightarrow c$ <br><br>$\therefore$ $\overrightarrow b .\overrightarrow c = 0$ <br><br>Let $\left| {\overrightarrow a + \vec b - \overrightarrow c } \right|$ = k <br><br>Square both sides <br><br>k² = ${{{\left( {\overrightarrow a } \right)}^2}}$ + ${{{\left( {\overrightarrow b } \right)}^2}}$ + ${{{\left( {\overrightarrow c } \right)}^2}}$ + $2\overrightarrow a .\overrightarrow b$ - $2\overrightarrow b .\overrightarrow c$ - $2\overrightarrow a .\overrightarrow c$ <br><br>$\Rightarrow$ k² = ${{{\left( {\overrightarrow a } \right)}^2}}$ + ${{{\left( {\overrightarrow b } \right)}^2}}$ + ${{{\left( {\overrightarrow c } \right)}^2}}$ <br><br>$\Rightarrow$ k² = 2² + 4² + 4² = 36 <br><br>$\Rightarrow$ k = 6