Understanding Right and Left Handed Systems in Vector Calculus

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In summary, the conversation discusses the relationship between right and left handed systems and how they are determined by the positions of the vectors within the system. The speaker also mentions the importance of understanding the definition of orientation-preserving in order to prove the statement.
  • #1
laminatedevildoll
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If a, b, c are right handed system, then so are b, c, a, and c, a, b. In this case, the vectors a, c, b and c, b, a and b, a, c are a left handed system.

In order to prove the above statement, I know that the right handed system is positive and the left handed system is negative. So for the second part, b,c,a and c, a, b are right handed because the positions were switched two times, so that makes a right handed system. But, a, c, b and c, b, a and b, a, c are only switched once, so that makes a left handed system. Is this how I prove this statement?
 
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  • #2
first you need a definition of right handed and left handed. what is it? or of orientation - preserving. then just check that these cases obey the definition.
 
  • #3
This is not a formal definition but, when the determinant is negative, it's left handed and if it's positive, it's right handed.
 

1. What is the difference between a right-handed and left-handed coordinate system?

A right-handed coordinate system is a mathematical system in which the direction of positive rotation is counterclockwise and the direction of positive translation is to the right. A left-handed coordinate system is the opposite, with positive rotation being clockwise and positive translation being to the left.

2. How are right-handed and left-handed coordinate systems used in science?

Right-handed and left-handed coordinate systems are used in various scientific fields such as physics, engineering, and mathematics. They are used to define the direction of forces, motion, and other physical quantities. They also play a crucial role in determining the orientation of objects in 3D space.

3. What is the significance of handedness in determining the direction of magnetic fields?

The direction of magnetic fields is determined by the right-hand rule, which is based on the concept of handedness. According to this rule, if you point your right thumb in the direction of the current flow, the direction in which your fingers curl represents the direction of the magnetic field.

4. Are there any real-world applications of right-handed and left-handed coordinate systems?

Yes, there are several real-world applications of right-handed and left-handed coordinate systems. For example, they are used in computer graphics to define the orientation of objects in 3D space, in navigation systems to determine the direction of travel, and in astronomy to describe the direction of rotation of celestial bodies.

5. Can a right-handed coordinate system be converted into a left-handed one?

Yes, a right-handed coordinate system can be converted into a left-handed one by simply reversing the direction of one of the axes. This is known as a mirror transformation and can be useful in certain applications where a left-handed coordinate system is preferred.

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