Rotation of axes
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Angles between lines through the origin
Let OP and OPī be two lines through the origin with direction cosines [Equation goes here - download the original pdf to see it.] respectively. Then the angle[symbol] between the two lines is given by [Equation goes here - download the original pdf to see it.] Proof. Firstly, the angle between two lines is the same regardless of how long the lines are, so we can specify that [Equation goes here - download the original pdf to see it.] which simplifies the proof. [Diagram goes here - download the original pdf to see it.] The coordinates of OP and OPī respectively are [Equation goes here - download the original pdf to see it.] By the cosine rule [Equation goes here - download the original pdf to see it.] Furthermore [Equation goes here - download the original pdf to see it.] Hence [Equation goes here - download the original pdf to see it.] This follows since [Equation goes here - download the original pdf to see it.] when [Equation goes here - download the original pdf to see it.] are direction angles. Example (4) Find the angle between the lines whose direction cosines are [Example goes here - download the original pdf to see it.] The orthogoanal projection of OP onto OQ is defined by the diagram [Diagram goes here - download the original pdf to see it.] and is given by [Equation goes here - download the original pdf to see it.] where [symbol] is the angle between OP and OQ. Result Let P be the point [Equation goes here - download the original pdf to see it.], then the orthogonal projection of OP onto OQ is given by [Equation goes here - download the original pdf to see it.] where [symbol]be the direction angles between OQ and the [Equation goes here - download the original pdf to see it.] axes respectively. [Diagram goes here - download the original pdf to see it.] Proof Let [Equation goes here - download the original pdf to see it.] The direction cosines of [Equation goes here - download the original pdf to see it.] are [Equation goes here - download the original pdf to see it.] Substituting into the formula [Equation goes here - download the original pdf to see it.] [Equation goes here - download the original pdf to see it.] The orthogonal projection of OP onto OQ is [Equation goes here - download the original pdf to see it.] whence [Equation goes here - download the original pdf to see it.] Example (5) Find the length of the orthogonal projection of the line OP where P is the point [Equation goes here - download the original pdf to see it.] and the line OQ where Q is the point [Equation goes here - download the original pdf to see it.]. [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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