The book states, "If the -axes are obtained from -axes by rotating through an angle , then and are related by either of the pair of equations:

How did the book obtain those equations?

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- Jan 7th 2011, 12:28 PMdwsmithFirst-order equations by introducing new coordinates.

The book states, "If the -axes are obtained from -axes by rotating through an angle , then and are related by either of the pair of equations:

How did the book obtain those equations? - Jan 7th 2011, 12:46 PMsnowtea
Draw the orthogonal axes for , and the rotated axes of by .

The equations are obtained by simple trig and vector addition. - Jan 7th 2011, 01:46 PMdwsmith
- Jan 7th 2011, 02:00 PMsnowtea
A drawing normally explains this very easily, but since I suck at drawing (and cannot seem to find a picture online for this common transformation), I will try to explain it analytically.

Any axis can be defined by a unit vector parallel to it in the postive direction.

So let be the unit vectors parallel to the corresponding axes.

A point in the coordinate system can be represented by the vector .

For any vector , its representation in the (orthogonal) coordinate system is:

.

So the coordinate is given by .

So for , this is .

Similarly for ...

[Edit: Found an image here: http://www.tutornext.com/system/file...Fig.1.36_0.GIF,

perhaps this will help with intuition using simple trigonometry] - Jan 7th 2011, 02:19 PMdwsmith
- Jan 7th 2011, 02:21 PMsnowtea
are unit vectors parallel to the axes. Their dot product is cosine of the angle between them (they have unit length).

- Jan 7th 2011, 02:24 PMdwsmith
Ok that makes since since but I have never seen sine defined in this manner so how does sine come into play?

- Jan 7th 2011, 02:36 PMsnowtea