The standard rules of Calculus apply for vector derivatives. It’s just that there is also a physical interpretation that must go along with it.

One of the most common examples of a vector derivative is angular acceleration, which is the derivative of the angular velocity vector. In other words, the angular acceleration vector is the rate of change of the angular velocity vector.

To help you visualize this, consider the figure below showing an angular velocity vector (

We can express

To calculate the angular acceleration vector, we calculate the difference in the angular velocity vector over a very small time step

Using calculus, the angular acceleration is calculated by taking the limit as

That’s all there is to it!

Now, there are situations (especially in two-dimensional problems) where it is not necessary to take the derivative of a vector, and the problem can be solved using techniques that are perhaps more familiar to the student.

This is illustrated with the following example. Let’s say a wheel of radius

To solve this problem we can express the position of the point

For the point

We set up the position of the particle

To find the velocity, take the first derivative of

Since

The point

The velocity of point

If we want to use the vector derivative approach to solve for the velocity of point

Set

where

Instant 1

Instant 2

As you can see, between instants 1 and 2 the position of point

We are now ready to differentiate the above two equations with respect to time:

At the given instant, the velocity at point

The term

Using calculus, and taking the limit as

Therefore,

As a result,

This is the same answer as before.

However, even though both approaches work, for three-dimensional kinematics and dynamics problems it is generally easier to evaluate vector derivatives, since that tends to make the solution as simple as possible (this becomes evident as one works through problems involving three-dimensional motion). In addition, equations have been specifically developed for rigid body motion, which involve vector derivatives. To see these equations click here.

Using the understanding gained thus far, we can derive a formula for the derivative of an arbitrary vector of fixed length in three-dimensional space.

Consider the figure below.

Where:

We wish to find an expression for

Since the vector

We can write:

Hence, using the vector cross product we have a very useful formula relating the derivative of a vector of fixed length to the angular velocity that "rotates" this vector, in three-dimensional space. Note that the above formula also applies if vector

Using the previous result we can derive a general formula for the derivative of an arbitrary vector of changing length in three-dimensional space.

First, set

where

Now, take the vector derivative of

Since

As a result,

This formula reduces to the formula given in the previous section if

The physical interpretation of

By now it should be more clear on why a vector derivative can be very useful. Fundamentally it's the same as a regular derivative, but with a physical interpretation included.

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