rotation of axes
Changing the orientation of the reference axes while maintaining the origin. The main reason for rotating axes is that a given equation is much simpler in the new coordinate system than it was in the original system.

When the original x- and y-axes rotate counterclockwise through an angle symbol theta, for any point P(x,y), the original coordinates (x,y) will become the new coordinates (x´,y´), and they are:

symbol spacex´ = x cossymbol theta + y sinsymbol theta
symbol spacey´ = -x sinsymbol theta + y cossymbol theta

To derive the equation in the new coordinates, we need to express original coordinates in the new coordinates:

symbol spacex = x´ cossymbol theta - y´ sinsymbol theta
symbol spacey = x´ sinsymbol theta + y´ cossymbol theta

For an example of rotation, consider a simple equation y = x + 21/2, which is a line. When the original x- and y-axes rotate counterclockwise through an angle of 45°, original coordinates can be expressed as:

symbol spacex = x´ cos45° - y´ sin45°
symbol spacey = x´ sin45° + y´ cos45°

Therefore,

symbol spacex = x´ (21/2/2) - y´ (21/2/2)
symbol spacey = x´ (21/2/2) + y´ (21/2/2)

Hence, the equation y = x + 21/2 becomes:

symbol spacex´ (21/2/2) + y´ (21/2/2) = x´ (21/2/2) - y´ (21/2/2) + 21/2

symbol spacey´ = 1

In the new coordinates, the equation is a line parallel to the x´-axis, +1 unit away from the x´-axis.


Related Term: translation of axes


 
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