Q. 9) The vector equation of XY-plane is

(A) $$\overrightarrow{\mathrm{r}} \cdot \hat{\mathrm{k}}=0$$

(B) $$\overrightarrow{\mathrm{r}} \cdot \hat{\mathrm{j}}=0$$

(C ) $$\overrightarrow{\mathrm{r}} \cdot \hat{\mathrm{i}}=0$$

(D) None of these

Solution:
Answer: (A) $$\vec{r} \cdot \hat{k}=0$$
The vector equation of the XY-plane is $$\vec{r} \cdot \hat{k}=0$$.

Explanation
– The XY-plane is a plane in 3D space where every point has a z -coordinate of zero, which corresponds to the Cartesian equation $$z=0$$.
– The normal vector (a vector perpendicular) to the XY-plane is the unit vector along the Z -axis, which is $$\hat{k}$$.
– The general vector equation of a plane that passes through the origin with a normal vector $$\vec{n}$$ is given by $$\vec{r} \cdot \vec{n}=0$$.
– Substituting $$\hat{k}$$ as the normal vector to the XY-plane results in the equation $$\vec{r} \cdot \hat{k}=0$$.