This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: E—102 Transforming Unit Vectors Between
Two Reference Frames Differing by a Single Angle Consider the two reference frames R and R1 with their respective unit vectors i, j, k and i1, j1 and k1 shown below. Their orientation differs only by the angle of rotation 8 , which can
be constant or a function of time. Assume we wish to express i, j and k in terms of i], jl, and
k1. ’ Y 2’2 u
‘ Fig. 1 Let us begin with the relation for i. We begin by breaking up the unit vector i into
components parallel to and perpendicular to the directions of i1 and jl as shown below. [i [cosG Fig. 2 Now, the component of i in the direction of i1 is simply lijcose. Since the component
is in the direction of i1, we can create a vector component as [Moose1‘1 = coseil. Now the component of i in the direction of j1 is simply lilsinB. Since the component is in the negative
direction of j I, we can create a vector component as ~ilsin6 j1 = —sin6 jl. Thus finally, i : costii1 ~ sinGj1 (1) Now the process described above can be time consuming There is, however a short cut.
Consider Figs 1. Let B : 0. Obviously, in this case, i = i1. The trig function which is maximum when 9 = O is cosine. Thus, one component for the case when 6 t 0 is coseil. Including the cosine automatically handles the component when 6 s 0. Now let 6 : 90° .
Obviously, in this case i = —jl. The trig function which is maximum when 8 = 90° is sine. Thus, one component for the case when 6 at 0 is —sir16 j1. Including the sine automatically handles the component when 6 at O. The final result is obtained by adding the two components
just obtained as i goosei1 — sinej1 (2)
the same result as given in Eq. 1‘ This procedure can be continued to express j in terms of i1 and jx. A little practice with
this technique will allow you to do it quickly and easily. ...
View Full
Document
 Winter '08
 Eke

Click to edit the document details