Point rotating in a coordinate system

• Karol
In summary, the coordinates of point P, denoted by x1 and x2, are rotated by angle α to point P', denoted by x'1 and x'2. It can be proven that x'1 equals x1 times the cosine of α plus x2 times the sine of α, and x'2 equals x2 times the cosine of α minus x1 times the sine of α. This can be shown by considering a vector with initial point at the origin and terminal point at P, and rotating it by α degrees.
Karol
The point P rotates with angle α to point P'. the coordinates of the old P are x1 and x2 and for P': x'1 and x'2.
Prove that:
$$x'_1=x_1\cos\alpha+x_2\sin\alpha$$
$$x'_2=x_2\cos\alpha-x_1\cos\alpha$$

I drew on the left the problem and on the right my attempt. the line OA, which is made of ##x_1\cos\alpha## plus ##x_2\sin\alpha## which is the blue line is indeed x'1 but i don't see the congruent triangles.

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• Snap1.jpg
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##P(x_1,x_2)## and If we rotate α degree we get new coordinates ##P'(x'_1,x'_2)##.Now Let make a vector which initial point Origin and terminal point P.This vector has magnitude R.Now we can show this vector in like this P=R(cosβ+sinβ) so ##x_1=Rcosβ## and ##x_2=Rsinβ##.Now we want to rotate this coordinate α degree.

and ##x'_2=Rsin(β+α)##.Think this way.

Thanks RyanH, i solved

1. What is a point rotating in a coordinate system?

A point rotating in a coordinate system refers to a point or object moving around a fixed origin in a specific direction and at a specific speed within a coordinate system. This movement can be described using mathematical equations and can help determine the position of the point at any given time.

2. How is the rotation of a point in a coordinate system calculated?

The rotation of a point in a coordinate system is calculated using trigonometry and the angle of rotation. The x and y coordinates of the point can be determined using the sine and cosine functions, which take into account the angle of rotation and the distance from the origin.

3. What is the significance of a point rotating in a coordinate system?

A point rotating in a coordinate system has many practical applications, such as in physics and engineering. It can be used to determine the position and movement of objects in space, as well as to create animations and simulate real-world scenarios.

4. Can a point rotate in a coordinate system in multiple dimensions?

Yes, a point can rotate in a coordinate system in multiple dimensions. The equations used to describe the rotation of a point in a 2D coordinate system can be extended to 3D and higher dimensions, allowing for more complex movements and calculations.

5. How does the rotation of a point in a coordinate system affect other points?

The rotation of a point in a coordinate system can affect the positions of other points if they are connected or dependent on each other. For example, in a rotating object, the movement of one point can cause a chain reaction of movements in other points. This can be visualized and analyzed using mathematical models and simulations.

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