Rotation of Axes: Point (x,y) to (X,Y)

In summary, if the coordinate axes are shifted clockwise by an angle \theta, the coordinates of the new point (X,Y) can be determined using trigonometric functions in relation to the original point (x,y) and the angle \theta.
  • #1
atavistic
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Let a point be(x,y)

If the coordinate axes are shifted in clockwise direction by an angle theeta,what are the coordinate of the new point(X,Y).
 
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  • #2
How about drawing a picture. Draw the x-axis horizontal, the y-axis vertical and mark the point (x,y). Now draw the x'-axis at angle [itex]\theta[/itex] to the x-axis, y'-axis at angle [itex]\theta[/itex] to the y-axis. Draw a perpendicular to the x-axis. Its length is y. Draw a perpendicular to the x'-axis. Its length is y'. You now have two right triangles and should be able to use trig functions to deternine y' as a function of x, y, and [itex]\theta[/itex].
 
  • #3


The new coordinates (X,Y) can be found by using the following transformation equations:

X = x*cos(theta) - y*sin(theta)
Y = x*sin(theta) + y*cos(theta)

These equations are derived from the rotation matrix for a 2D coordinate system. The angle theta represents the amount of rotation in radians.

To better understand this transformation, imagine a point (x,y) on a graph. If we rotate the axes by an angle theta, the point (x,y) will also rotate by the same amount, resulting in the new coordinates (X,Y). This rotation can be visualized as a rotation of the entire coordinate system, with the point (x,y) remaining fixed in place.

This transformation, known as the rotation of axes, is commonly used in mathematics and science to simplify equations and solve problems involving rotation. It is also used in computer graphics to rotate images and objects.

In summary, when the coordinate axes are rotated by an angle theta, the new coordinates (X,Y) of a point (x,y) can be found using the transformation equations X = x*cos(theta) - y*sin(theta) and Y = x*sin(theta) + y*cos(theta). This concept is important in various fields of science and mathematics, and understanding it can help in solving many problems involving rotation.
 

1. What is the concept of "rotation of axes" in mathematics?

The concept of rotation of axes is a transformation in which the coordinate axes are rotated about the origin by a certain angle. This transformation is used to represent the same point in a different coordinate system.

2. How is a point (x,y) transformed to (X,Y) using rotation of axes?

To transform a point (x,y) to (X,Y) using rotation of axes, we first determine the angle of rotation and then use the following formulas:
X = x*cosθ - y*sinθ
Y = x*sinθ + y*cosθ
where θ is the angle of rotation.

3. What is the purpose of using rotation of axes?

The purpose of using rotation of axes is to simplify the representation of geometric figures and equations. It allows us to change the orientation of the coordinate system and make it easier to solve problems involving angles and rotations.

4. Is it possible to rotate the axes by any angle?

Yes, it is possible to rotate the axes by any angle. The angle of rotation can be positive or negative, and it can be any value between 0 and 360 degrees.

5. Can rotation of axes be applied to three-dimensional coordinates?

Yes, rotation of axes can also be applied to three-dimensional coordinates. In this case, we use three angles of rotation (one for each axis) to determine the new coordinates of a point in the rotated system.

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