In the study of rotational dynamics, understanding the direction of vectors like angular momentum is crucial, especially in the context of orbital motion. The question asks about the direction of the angular momentum vector in an orbital motion.

Angular Momentum Vector Direction

The angular momentum vector, denoted as $\vec{L}$, for a particle in motion about a fixed point is defined by the cross product of its position vector $\vec{r}$ (from the origin to the particle) and its linear momentum vector $\vec{p}$. Mathematically, this is expressed as:

$$\vec{L} = \vec{r} \times \vec{p}$$

Where:

• $\vec{r}$ is the position vector (also known as the radius vector) from the center of rotation to the particle.
• $\vec{p}$ is the linear momentum vector of the particle, given by $\vec{p} = m\vec{v}$ (mass times velocity).

Orbital Motion Context

In orbital motion, such as a planet orbiting a star or an electron orbiting a nucleus, the movement typically occurs within a well-defined plane, which is known as the orbital plane. Both the position vector $\vec{r}$ and the linear momentum vector $\vec{p}$ (which is always tangent to the path of motion, and thus lies within the orbital plane) are contained within this orbital plane.

Vector Cross Product Explanation

The fundamental property of a vector cross product is that the resulting vector is always perpendicular to the plane formed by the two original vectors. The direction of the cross product $\vec{A} \times \vec{B}$ is determined using the right-hand rule. If you point the fingers of your right hand in the direction of the first vector ($\vec{r}$) and then curl them towards the direction of the second vector ($\vec{p}$), your thumb will indicate the direction of the resultant angular momentum vector ($\vec{L}$).

Since both the position vector $\vec{r}$ and the linear momentum vector $\vec{p}$ lie entirely within the orbital plane, their cross product, the angular momentum vector $\vec{L}$, must necessarily be perpendicular to that same orbital plane.

Evaluating the Options

• Along the radius vector: This is incorrect. The angular momentum vector is perpendicular to the radius vector, not aligned with it.
• Parallel to the linear momentum: This is also incorrect. The angular momentum vector is perpendicular to the linear momentum vector.
• In the orbital plane: This is incorrect. As explained by the properties of the cross product, the angular momentum vector is directed perpendicular to the orbital plane, meaning it does not lie within the plane itself.
• Perpendicular to the orbital plane: This statement accurately describes the direction of the angular momentum vector in orbital motion. This direction is a direct consequence of the cross product relationship between the position and linear momentum vectors, both of which reside within the orbital plane.

Therefore, in an orbital motion, the angular momentum vector is always perpendicular to the orbital plane.