Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub

Projection of $\bar{v}$ along $\bar{u}$ and $\bar{w}$ along $\bar{u}$ is $\frac{\bar{v}\cdot \bar{u}}{|\bar{u}|}$ and $\frac{\bar{w}\cdot \bar{u}}{|\bar{u}|}$ respectively According to question $\frac{\overline{\mathrm{v}}\cdot \overline{\mathrm{u}}}{|\overline{\mathrm{u}}|}=\frac{\overline{\mathrm{w}}\cdot \overline{\mathrm{u}}}{|\overline{\mathrm{u}}|}\Rightarrow \overline{\mathrm{v}}\cdot \overline{\mathrm{u}}=\overline{\mathrm{w}}\cdot \overline{\mathrm{u}}$. and $\overline{\mathrm{v}}\cdot \overline{\mathrm{w}}=0$ $$|\bar{u}-\bar{v}+\bar{w}{|}^2=|\bar{u}{|}^2+|\bar{v}{|}^2+|\bar{w}{|}^2-2\bar{u}\cdot \bar{v}+2\bar{u}\cdot \bar{w}-2\bar{v}\cdot \bar{w}=14\Rightarrow |\bar{u}-\bar{v}+\bar{w}|=\sqrt{14}$$