Let \(\bar{u}=\hat{i}+\hat{j}, \bar{v}=\hat{i}-\hat{j}\) and \(\bar{w}=\hat{i}+2 \hat{j}+3 \hat{k}\). If \(\hat{n}\) is a unit vector such that \(\overline{\mathrm{u}} \cdot \hat{\mathrm{n}}=0\) and \(\overline{\mathrm{v}} \cdot \hat{\mathrm{n}}=0\), then \(|\overline{\mathrm{w}} \cdot \hat{\mathrm{n}}|\) is equal to

We have, \(\overline{\mathrm{u}} \cdot \hat{\mathrm{n}}=0\) and \(\overline{\mathrm{v}} \cdot \hat{\mathrm{n}}=0\)
\(\therefore \quad \hat{\mathrm{n}} \perp \overline{\mathrm{u}}\) and \(\hat{\mathrm{n}} \perp \stackrel{\rightharpoonup}{\mathrm{v}}\)
\(\Rightarrow \hat{\mathrm{n}}= \pm \frac{\overline{\mathrm{u}} \times \overline{\mathrm{v}}}{|\overline{\mathrm{u}} \times \overline{\mathrm{v}}|}\)
Now, \(\bar{u} \times \bar{v}=(\hat{i}+\hat{j}) \times(\hat{i}-\hat{j})=-2 \hat{k}\)
\(\therefore \quad \hat{\mathrm{n}}= \pm \hat{\mathrm{k}}\)
Hence, \(|\overline{\mathrm{w}} \cdot \hat{\mathrm{n}}|=|(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}) \cdot( \pm \hat{\mathrm{k}})|=3\)