Rotation of axes
DOWNLOAD
FREE
|
Rotation of Axes - Introduction
In geometry we represent a point P in space by coordinates [Equation goes here - download the original pdf to see it.]; the coordinates are referred to a basis of vectors that act as the axes. Here we assume that the axes are also orthonormal, meaning, a right-angles to each other. The choice of axes are somewhat arbitrary, and it is appropriate to ask what would be the coordinates of the same point P if another set of axes were chosen? In this chapter we are specifically concerned with the rotation of a set of axes, and we leave the possibility of a translation of the origin to another time. Let the coordinates of P relative to this second orthonormal set of axes be [Equation goes here - download the original pdf to see it.]. We are seeking a process for transforming the point [Equation goes here - download the original pdf to see it.] to [Equation goes here - download the original pdf to see it.]; a process that gives [Equation goes here - download the original pdf to see it.] in terms of [Equation goes here - download the original pdf to see it.]. By this stage you should be familiar with the idea that square matrices specify transformations of space - they describe rotations, reflections, skews, enlargements, dilations and so forth - so it would be natural to expect the answer to this present question to result in a square matrix that will be the transformation matrix of rotation of axes, and this is indeed the case, as we proceed to demonstrate.
|
Contents of
Rotation of axes
1 Rotation of Axes - Introduction
2 Right and left-hand coordinates
3 Direction cosines
4 Angles between lines through the origin
5 Rotation of axes
6 The summation convention
7 Invariance with respect to a rotation of the axes
|
