To determine the correct representation of the position vector [tex]\( \vec{r} \)[/tex] of a point in the [tex]\( xz \)[/tex]-plane, let's examine each option and the given plane.
The [tex]\( xz \)[/tex]-plane is defined by the fact that any point on this plane has its [tex]\( y \)[/tex]-coordinate equal to zero. This matches our intuitive understanding that the plane formed by the [tex]\( x \)[/tex] and [tex]\( z \)[/tex] axes does not involve any displacement in the [tex]\( y \)[/tex] direction.
Let's analyze each option:
- Option A: [tex]\( \vec{r} = x \hat{i} + y \hat{j} \)[/tex]
In this expression, the vector [tex]\( \vec{r} \)[/tex] has both [tex]\( x \)[/tex] and [tex]\( y \)[/tex] components. Since points on the [tex]\( xz \)[/tex]-plane have no [tex]\( y \)[/tex]-coordinate, this representation cannot be correct.
- Option B: [tex]\( \vec{r} = y \hat{i} + z \hat{k} \)[/tex]
Here, the vector [tex]\( \vec{r} \)[/tex] has components in the [tex]\( y \)[/tex]- and [tex]\( z \)[/tex]-directions. However, in the [tex]\( xz \)[/tex]-plane, positions are not defined by [tex]\( y \)[/tex]-coordinates. Therefore, this option is also incorrect.
- Option C: [tex]\( \vec{r} = x \hat{i} + y \hat{j} + z \hat{k} \)[/tex]
This option includes components in all three directions: [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex]. Since we are specifically looking for the position vector in the [tex]\( xz \)[/tex]-plane where [tex]\( y \)[/tex] is always zero, this expression is too general.
- Option D: [tex]\( \vec{r} = x \hat{i} + z \hat{k} \)[/tex]
Finally, this option gives the position vector in terms of just the [tex]\( x \)[/tex] and [tex]\( z \)[/tex] components, with no [tex]\( y \)[/tex]-component. This correctly represents the position of a point in the [tex]\( xz \)[/tex]-plane.
Given the constraints of the [tex]\( xz \)[/tex]-plane, Option D: [tex]\( \vec{r} = x \hat{i} + z \hat{k} \)[/tex] correctly and succinctly represents the position vector of a point on this plane.
Thus, the correct answer is:
[tex]\[ \vec{r} = x \hat{i} + z \hat{k} \][/tex]
