Physics - Practical physics Question with Solution | TestHub

$Y=A\sin (\omega t-\vec{k}\cdot \vec{r})$

$$\begin{array}{rl}\vec{r}& =x\hat{i}+y\hat{j}+z\hat{k}\\ \vec{k}\cdot \vec{r}& =(3x+2y-z)\pi \\ \vec{k}& =(3\hat{i}+2\hat{j}-\hat{k})\pi \\ |\vec{k}|& =\pi \sqrt{9+4+1}=\pi \sqrt{14}\\ |\vec{k}|& =\frac{2\pi }{\lambda }\Rightarrow \lambda =\frac{2\pi }{|\vec{k}|}=\frac{2\pi }{\pi \sqrt{14}}=\frac{2}{\sqrt{14}}=\sqrt{\frac{4}{14}}=\sqrt{\frac{2}{7}}m\end{array}$$

The unit vector in the direction of wave propagation is $\frac{3\hat{i}+2\hat{j}-\hat{k}}{\sqrt{14}}$.