Question

If $\overrightarrow{a},\overrightarrow{b}\mathrm{and}\overrightarrow{c}$are mutually perpendicular unit vectors, write the value of $\left|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\right|.$

Answer 1

$$\begin{gathered} \text{Given that}\overrightarrow{a}\text{,}\overrightarrow{b}\text{and}\overrightarrow{c}\text{are unit vectors.} \\ \text{So,}\left|\overrightarrow{a}\right|\text{=1,}\left|\overrightarrow{b}\right|\text{=1 and}\left|\overrightarrow{c}\right|\text{=1 ...}\left(\mathrm{i}\right) \\ \text{Since they are mutually perpendicular,} \\ \overrightarrow{a}.\overrightarrow{b}=\overrightarrow{b}.\overrightarrow{c}=\overrightarrow{c}.\overrightarrow{a}=0...\left(\mathrm{ii}\right) \\ \text{Now,} \\ {\left|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\right|}^2={\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2+{\left|\overrightarrow{c}\right|}^2+2\overrightarrow{a}.\overrightarrow{b}+2\overrightarrow{b}.\overrightarrow{c}+2\overrightarrow{c}.\overrightarrow{a} \\ =1+1+1+0+0+0\left[\mathrm{Using}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right)\right] \\ =3 \\ \therefore \left|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\right|=\sqrt{3} \end{gathered}$$