Engineering Mechanics | Derivative of a vector

Derivative of a vector: After reviewing the vector algebra, we would now like to introduce you to the idea of differentiating a vector quantity. Here we take a vector clip_image001 as depending on one parameter, say time t , and evaluate the derivative clip_image002. This is similar to what we do for a regular function. We evaluate the vector clip_image003at time (t+ Δt) , subtract clip_image001[1] from it, divide the difference clip_image004 by Δt and then take the limit Δ t → 0 .

This is shown in figure 14. Thus



The derivative is easily understood if we think in terms of its derivatives. If we write a vector as


then the derivative of the vector is given as


Notice that only the components are differentiated, because the unit vectors clip_image009are fixed in space and therefore do not change with time. Later when we learn about polar coordinates, we will encounter unit vectors which also change with time. In that case when taking derivative of a vector, the components as well as the unit vectors both have to be differentiated.

Using the definition above, it is easy to show that in differentiating the product of two vectors, the usual chain rule can be applied. This gives




This pretty much sums up our introduction to vectors. I leave this lecture by giving you three exercises.

1. Show that clip_image012and that clip_image013is the volume of a parallelepiped formed by clip_image014.

2. Show that clip_image015can also be written as the determinant


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