Understanding the Dot Product of Derivatives in Astrodynamics

In summary, the statement \vec{a}\cdot \vec{\dot{a}}=a \dot{a} indicates that the dot product between the vector \vec{a} and its time derivative \vec{\dot{a}} is equal to the product of the magnitude of \vec{a} and the time derivative of its magnitude. This can be seen as a notation for the time derivative of the magnitude of a vector. The statement can be visually referenced in the provided image. The statement may initially seem counterintuitive, but upon further examination, it can be understood as a notation for the time derivative of a vector's magnitude.
  • #1
Nabeshin
Science Advisor
2,207
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In reading a book on astrodynamics I came across the following statement:
[tex]\vec{a}\cdot \vec{\dot{a}}=a \dot{a}[/tex]

Where the dotting is the time derivative notation.

I put a picture of the original text up, and it's the statement right in the middle:
http://img.photobucket.com/albums/v715/deagleman9/IMG_1178.jpg
Except they use bold to indicate vectors.

Can anyone explain to me why this should be true? It seems akin to saying the angle between a vector and its time derivative is always 90, which is obviously not true. I've also considered it might be a notational problem with the unbolded quantities. At any rate, does anyone know what's going on here?
 
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  • #2
Never mind, I think I figured this one out. I think the right side is the time derivative of the magnitude of a, rather than the magnitude of the time derivative of a.
 

FAQ: Understanding the Dot Product of Derivatives in Astrodynamics

What is the dot product of derivatives?

The dot product of derivatives is a mathematical operation that involves taking the derivatives of two functions and multiplying them together, then adding up the results. It is used to calculate the rate of change of a vector quantity in a particular direction.

How is the dot product of derivatives calculated?

To calculate the dot product of derivatives, you first take the derivatives of the two functions involved. Then, you multiply the derivatives together term by term and add up the results. The resulting expression is the dot product of the derivatives.

What is the geometric interpretation of the dot product of derivatives?

The dot product of derivatives has a geometric interpretation as the projection of one vector onto another. It represents the magnitude of the change in the direction of the vector quantity as it moves in the direction of the other vector.

What are some real-world applications of the dot product of derivatives?

The dot product of derivatives has many applications in physics, engineering, and economics. It is used to calculate the work done by a force, the rate of change of temperature in a given direction, and the rate of change of a stock's value over time.

Can the dot product of derivatives be negative?

Yes, the dot product of derivatives can be negative. This happens when the two vectors involved have opposite directions, resulting in a negative projection. It can also be negative if the rate of change in one direction is decreasing while the other is increasing.

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