How does rotating a coordinate system affect vector direction and components?

In summary, when a coordinate system is moved, the vector's direction remains the same, but its components may change. This is because the vector is not physically affected, only the coordinate system is changed. This can be seen in the example of a compass needle pointing North regardless of how the compass is rotated. However, a coordinate system is still necessary for an unambiguous description of the vector.
  • #1
Mathematicsresear
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If I move a coordinate system by an angle theta, why does the vector still have the same direction, but the components are different?
 
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  • #2
Because you just changed the coordinate system, you did not change anything physically. If the vector pointed towards the Moon before it will still point towards the Moon if you pick another coordinate system.
 
  • #3
I had another example with a compass needle pointing North regardless of how you rotate the casing of the compass on which N-S & E-W markings are drawn, but @Orodruin got there first with his example. I should add, though, that this example shows that a coordinate system is always needed before an unambiguous description of a vector can be written down.
 
  • #4
kuruman said:
I should add, though, that this example shows that a coordinate system is always needed before an unambiguous description of a vector can be written down.
This is not really true. You just need a single reference direction in which the vector is pointing. For example, "the rocket is moving with a velocity of 0.5 towards the Earth".
 

1. What is coordinate system rotation?

Coordinate system rotation is a mathematical process used to transform the coordinates of a point or object from one coordinate system to another. This is often done to simplify calculations or to align the coordinates with a specific reference frame.

2. Why is coordinate system rotation important in science?

Coordinate system rotation is important in science because it allows researchers to analyze and compare data from different sources or experiments. It also helps to accurately describe and understand the position and movement of objects in space.

3. How is coordinate system rotation performed?

Coordinate system rotation is performed using a series of mathematical equations and transformations. These equations involve rotating the axes of the coordinate system and translating the coordinates to the new system.

4. What are the different types of coordinate system rotation?

There are three main types of coordinate system rotation: 2D rotation, 3D rotation, and compound rotation. 2D rotation involves rotating coordinates in a two-dimensional plane, while 3D rotation involves rotating coordinates in a three-dimensional space. Compound rotation combines multiple rotations to achieve a desired orientation.

5. What are some applications of coordinate system rotation in science?

Coordinate system rotation is used in various fields of science, such as astronomy, geology, and physics. It is used to track the movement of celestial bodies, map geological features, and analyze the motion of particles in physics experiments. It is also used in computer graphics and simulations to create realistic images and animations.

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