Physical Interpretation of a Vector Quantity

[Bashyboy]

Homework Statement

If $\mathbf{r}, \mathbf{v},\mathbf{a}$ denote the position, velocity, and acceleration of a particle, prove that

$\frac{d}{dt} [\mathbf{a} \cdot (\mathbf{v} \times \mathbf{r})] = \dot{ \mathbf{a}} \cdot (\mathbf{v} \times \mathbf{r})$

Homework Equations

The Attempt at a Solution

I have already proven the result, but am now wondering the what the physical significance of this vector quantity is. I believe I have some idea:

The vectors $\mathbf{v}$ and $\mathbf{r}$ create their own two dimensional subspace (a plane) in $\mathbb{R}^3$, with $\mathbf{v} \times \mathbf{r}$ being normal to this plane. The dot product of this with the acceleration vector $\mathbf{a}$ gives the projection of the acceleration vector onto the normal vector, it is a measure of how parallel they are. If the dot product is zero, then this implies $\mathbf{a}$ is orthogonal to the vector $\mathbf{v} \times \mathbf{r}$, and that it lies in the plane $\mathbf{v}$ and $\mathbf{r}$ create. If it is not zero, then the this implies that the acceleration vector has a component lying in the plane, and a component parallel to the normal of the plane. This causes the particle to spiral away from the plane.

Does this seem correct?

 

[AlephZero]

Can you define your notation better?

The question says $\mathbf{a}$ denotes the position. In your solution you say it is the acceleration vector.

What do $\mathbf{b}$ and $\mathbf{c}$ have to do with the question?

And what are $\mathbf{v}$ and $\mathbf{r}$?

 

[Bashyboy]

Whoops, I am terribly sorry. I just edited my original post.

 

[AlephZero]

Thanks. In your first version of the question, I wasn't sure if v and r were supposed to be a rotating coordinate system or something similar.

For the geometrical interpretation, See http://en.wikipedia.org/wiki/Frenet–Serret_formulas

 

[paranormal]

Yes, your interpretation is correct. The quantity \mathbf{a} \cdot (\mathbf{v} \times \mathbf{r}) is known as the "angular velocity" of the particle, and it represents the rate at which the particle is changing direction in its motion. This can be visualized as the particle rotating around the axis defined by \mathbf{v} \times \mathbf{r}. The dot product with \mathbf{a} then represents the component of the acceleration that is tangential to this rotation, while the cross product \mathbf{v} \times \mathbf{r} represents the normal component. This is a useful concept in understanding the motion of objects in three-dimensional space, and can be applied to a variety of physical systems, such as rotating objects or particles in circular motion.