Rotation of axes
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Rotation of axes
Consider two sets of coordinate axes, [Equation goes here - download the original pdf to see it.] and [Equation goes here - download the original pdf to see it.]. If one set is right-handed and the other is left-handed, then it is impossible to rotate the one so that it coincides with the other. So let us suppose that both are right-handed coordinate axes. [Diagram goes here - download the original pdf to see it.] The coordinates of a point [Equation goes here - download the original pdf to see it.] relative to the coordinates [Equation goes here - download the original pdf to see it. are [Equation goes here - download the original pdf to see it.]. We wish to find the coordinates of P relative to the coordinates [Equation goes here - download the original pdf to see it.] Now the rotation is given by nine angles, but there are relationships between these angles, since the rotated set of axes [Equation goes here - download the original pdf to see it.] remains orthogonal (i.e. at right-angles to each other). [Diagram goes here - download the original pdf to see it.] The diagram above shows three of these angles, those made by the Ox axis with the [Equation goes here - download the original pdf to see it.] respectively, which we have labelled [Equation goes here - download the original pdf to see it.] respectively. The coordinates of the point P relative to [Equation goes here - download the original pdf to see it.] are just [Equation goes here - download the original pdf to see it.] but by the result [Equation goes here - download the original pdf to see it.] we have [Equation goes here - download the original pdf to see it.] Similarly, if [Equation goes here - download the original pdf to see it.] are the angles made by the y-axis with [Equation goes here - download the original pdf to see it.] respectively, and [Equation goes here - download the original pdf to see it.] are the angles made by the z-axis with [Equation goes here - download the original pdf to see it.] respectively, the [Equation goes here - download the original pdf to see it.] Putting [equation] and so forth, we obtain a matrix representation for transformation of the coordinates of P from the coordinates [Equation goes here - download the original pdf to see it.] to [Equation goes here - download the original pdf to see it.] as [Equation goes here - download the original pdf to see it.] The matrix [Equation goes here - download the original pdf to see it.] is called the transformation matrix of the rotation of axes. As indicated above, the fact that both sets of coordinates are orthonormal imposes conditions on the structure of the matrix A. Two lines are perpendicular when the cosine of the angle between them is 0; that is, if [Equation goes here - download the original pdf to see it.] is the angle between them, then [Equation goes here - download the original pdf to see it.]. So [Equation goes here - download the original pdf to see it.] Now, [Equation goes here - download the original pdf to see it.] is an orthonormal set, so [Equation goes here - download the original pdf to see it.] Since the sum of the squares of direction cosines is 1 [Equation goes here - download the original pdf to see it.] These six relationships are called the orthonormality conditions. The transformation matrix of [Equation goes here - download the original pdf to see it.] relative to [Equation goes here - download the original pdf to see it.]i s [Equation goes here - download the original pdf to see it.] Here the matrix [Equation goes here - download the original pdf to see it.] is the transpose of A; that is, writing [Equation goes here - download the original pdf to see it.] [Equation goes here - download the original pdf to see it.] then [Equation goes here - download the original pdf to see it.] The product [Equation goes here - download the original pdf to see it.] gives the orthonormality conditions, that is [Equation goes here - download the original pdf to see it.] so [Equation goes here - download the original pdf to see it.] However, we can show that [Equation goes here - download the original pdf to see it.] for all transformation matrices, for firstly, if the two coordinate systems coincide, then [Equation goes here - download the original pdf to see it.]and [Equation goes here - download the original pdf to see it.]. Then, if the axes of [Equation goes here - download the original pdf to see it.] are rotated then the direction cosines will vary continuously; likewise, the determinant will vary continuously, which means the determinant cannot jump discretely from +1 to -1; hence [Equation goes here - download the original pdf to see it.] for all transformation matrices. This also confirms that if [Equation goes here - download the original pdf to see it.] is a set of right-handed coordinates, then, if [Equation goes here - download the original pdf to see it.] where A is a transformation matrix, then[Equation goes here - download the original pdf to see it.] is also right-handed. Example (6) Find the transformation matrix when a set of axes is rotated clockwise (i.e. from the x-axis to the y-axis) through an angle of [Equation goes here - download the original pdf to see it.] about the z-axis. [Example goes here - download the original pdf to see it.] Example (7) Show that the following [Example goes here - download the original pdf to see it.]
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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
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