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Fundamentals of Stress and Vibration 1. Mathematics for Structural mechanics
[A Practical guide for aspiring Designers / Analysts]

In order to get ‘y’ in terms of ‘y ’, let us multiply equation (1.45) by ‘−sinα’ and equation (1.46) by
1
‘cosα’ and add the resulting equations.
−x sinα = −x sinα cosα + y sin α - - - - (1.51)
2
1
y cosα = y cos α + x cosα sinα - - - - (1.52)
2
1
Adding equations (1.51) and (1.52), we get:

2
2
−x sinα + y cosα = −x sinα cosα + y sin α + y cos α + x cosα sinα
1
1
−x sinα + y cosα = y cos α + sin α = −x sinα + y cosα = y - - - - (1.53)
2
2
1
1
1
1
Equations (1.50) and (1.53) can be rewritten in the matrix form to get a relationship between
original and transformed coordinates of vector V .

cosα sinα x 1 x
= - - - - (1.54)
−sinα cosα y 1 y
transformation matrix transformed vector = original vector
Let us understand the nature of the transformation matrices in equations (1.47) and (1.54).

It can be observed that the transformation matrix in equation (1.54) is the transpose or inverse of
the transformation matrix in equation (1.47).

cosα −sinα
Let the transformation matrix in equation . be ‘A’: = A
sinα cosα

cosα sinα
Let the transformation matrix in equation . be ‘B’: = B
−sinα cosα
We could now conclude the following:

1) A = B and B = A
T
T
2) B B = I (identity matrix) and (A A = I(identity matrix))
T
T

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QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries, Page 49