Let the vectors \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) be such that \(|\overline{\mathrm{a}}|=2,|\dot{\bar{b}}|=4\) and \(|\overline{\mathrm{c}}|=4\). If the projection of \(\overline{\mathrm{b}}\) on \(\overline{\mathrm{a}}\) is equal to the projection of \(\overline{\mathrm{c}}\) on \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{b}}\) is perpendicular to \(\overline{\mathrm{c}}\), then the value of \(|\overline{\mathrm{a}}+\overline{\mathrm{b}}-\overline{\bar{c}}|\) is equal to

\(||\bar{a}|=2,|\bar{b}|=4 \text { and }| \bar{c} \mid=4\)
According to the given condition, \((\) Projection of \(\bar{b}\) on \(\bar{a})=(\) Projection of \(\bar{c}\) on \(\bar{a}\) )
\(\begin{array}{ll}
\therefore & \frac{\overline{\mathrm{b}} \cdot \overline{\mathrm{a}}}{|\overrightarrow{\mathrm{a}}|}=\frac{\overline{\mathrm{c}} \cdot \overline{\mathrm{a}}}{|\overline{\mathrm{a}}|} \\
\therefore & \overline{\mathrm{b}} \cdot \overline{\mathrm{a}}=\overline{\mathrm{c}} \cdot \overline{\mathrm{a}} \\
\therefore & (\overline{\mathrm{~b}}-\overline{\mathrm{c}}) \cdot \overline{\mathrm{a}}=0
...(i)\end{array}\)
Now consider, \(|\bar{a}+\bar{b}-\bar{c}|\)
\(\begin{aligned}
& =\sqrt{|\bar{a}+\bar{b}-\bar{c}|^2} \\
& =\sqrt{|\bar{a}|^2+|\bar{b}-\bar{c}|^2+2 \bar{a} \cdot(\bar{b}-\bar{c})} \\
& =\sqrt{(2)^2+|\bar{b}-\bar{c}|^2+0}
\end{aligned}\)...[from(ii)]
\(\begin{aligned}
& =\sqrt{4+|\overline{\mathrm{b}}|^2+|\overline{\mathrm{c}}|^2-2(\overline{\mathrm{~b}} \cdot \overline{\mathrm{c}})} \\
& =\sqrt{4+(4)^2+(4)^2+0}
\end{aligned}\)
\(\ldots[\overline{\mathrm{b}}\) and \(\overline{\mathrm{c}}\) are perpendicular \(]\)
\(\begin{aligned}
& =\sqrt{36} \\
& =6
\end{aligned}\)