$\overrightarrow{\mathrm{u}}=2\hat{\mathrm{i}}+3\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{v}}=-3\hat{\mathrm{i}}+2\hat{\mathrm{j}}$, $\overrightarrow{\mathrm{w}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+4\hat{\mathrm{k}}$

$\overrightarrow{u}.\overrightarrow{v}=\left(2\hat{\mathrm{i}}+3\hat{\mathrm{j}}+\hat{\mathrm{k}}\right).\left(-3\hat{\mathrm{i}}+2\hat{\mathrm{j}}\right)$

$\Rightarrow \overrightarrow{u}.\overrightarrow{v}=-6+6+0$

$\Rightarrow \overrightarrow{u}.\overrightarrow{v}=0$ 

Therefore. $\overrightarrow{u}$ is perpendicular to $\overrightarrow{v}$.

Now, $\overrightarrow{u}.\overrightarrow{w}=(2\hat{i}.\hat{i})+(3\hat{j}.-\hat{j})+(\hat{k}.4\hat{k})$

$\Rightarrow \overrightarrow{u}.\overrightarrow{w}=2-3+4=3$

Therefore, $\overrightarrow{u}$ is not perpendicular $\overrightarrow{w}$.

Now, $\overrightarrow{\mathrm{w}}.\overrightarrow{v}=\left(\hat{\mathrm{i}}-\hat{\mathrm{j}}+4\hat{\mathrm{k}}\right).\left(-3\hat{\mathrm{i}}+2\hat{\mathrm{j}}\right)$

$\Rightarrow \overrightarrow{\mathrm{w}}.\overrightarrow{v}=-3-2+0=-5$

Therefore $\overrightarrow{w}$ is not perpendicular $\overrightarrow{v}$.