Page 44 - Introduction to Continuum Mechanics

Part B Transformation Laws for Cartesian Components of Vectors 29

or

Equation (2B12.1) is the transformation law relating components of the same vector with
respect to different rectangular Cartesian unit bases. It is very important to note that in
Eq. (2B12.1c), [a]' denote the matrix of the vector a with respect to the primed basis e,' and
[a] denote that with respect to the unprimed basis e/. Eq. (2B12.1) is not the same as
7*
a'=Q a. The distinction is that [a] and [a]' are matrices of the same vector, where a and a' are
T
two different vectors; a' being the transformed vector of a (through the transformation Q ).
If we premultiply Eq. (2B12.1c) with [Q], we get

The indicial notation equation for Eq.(2B12.2a) is

Example 2B 12.1
Given that the components of a vector a with respect to {e/} are given by (2,0,0), (i.e.,
a = 2ei), find its components with respect to {e/}, where the e/ axes are obtained by a 90°
counter-clockwise rotation of the e/ axes about the C3-axis.
Solution. The answer to the question is obvious from Fig. 2B.5, that is

a = 2ex = -2ei
We can also obtain the answer by using Eq. (2B12.2a). First we find the transformation matrix.
With e'i - e 2, ej> = -*i and €3 = e 3, by Eq. (2Bll.lb), we have

0 -1 0"
[Q]= 1 0 0
0 0 1
Thus,
I" 0 1 O] |~2] To "
[a]' = [Q] [a]= -10 0 0 = -2
001 0 0
i.e.,
a = -2e' 2