# Derive relations for components by rotation of axes

• briteliner
In summary, the conversation is about deriving the relations for the components of a 2 dimensional vector when referred to new axes obtained by rotating the original axes through an angle Θ. The equations for the new components (x1',x2') are x1'= cosΘ x1+sinΘ x2 and x2'=-sinΘ x1+cosΘ x2. The attempt at a solution involves using trigonometry to determine the components (1,0) and (0,1) when rotated by an angle Θ, and then using those components to find the new components of the vector. However, it is not clear if this solution is sufficient.
briteliner

## Homework Statement

x1,x2) are the components of a 2 dimensional vector r when referred to cartesian axes along the directions i,j. derive the relations
x1'= cosΘ x1+sinΘ x2
x2'=-sinΘ x1+cosΘ x2
for the components (x1',x2') or r referred to new axes i',j' obtained by a rotation of the axes through an angle Θ about the k direction

## The Attempt at a Solution

i just wrote that r=x^2+y^2 which in this case would be x1^2+x2^2 and then accounted for rotation by multiplying by sin or cos theta and proving that x1^2+x2^2=x1'^2+x2'^2 but it don't think its sufficient

What's (1,0) rotated by an angle theta? Ditto for (0,1). To get those just draw a right triangle in the xy plane with angle theta at the origin. Use trig. Then (x1,x2)=x1*(1,0)+x2*(0,1).

## 1. What is the purpose of deriving relations for components by rotation of axes?

The purpose of deriving relations for components by rotation of axes is to simplify the calculation of physical quantities in a different coordinate system. This is particularly useful in situations where the original coordinate system may be difficult to work with, but a rotated coordinate system would make the calculations easier.

## 2. How do you derive relations for components by rotation of axes?

The process of deriving relations for components by rotation of axes involves using the trigonometric functions sine and cosine to relate the components in the original coordinate system to the components in the rotated coordinate system. This is typically done by drawing a diagram and using the angle of rotation to determine the appropriate trigonometric functions to use.

## 3. What are the advantages of using rotated coordinate systems?

Using rotated coordinate systems can have several advantages, including simplifying complex calculations, making physical concepts easier to visualize, and reducing the number of variables needed to describe a system. It can also make it easier to solve problems in situations where there is symmetry or certain relationships between the components.

## 4. Are there any limitations to using rotated coordinate systems?

While rotated coordinate systems can be extremely useful, they are not always necessary or practical. In some cases, the calculations may be more complex in a rotated coordinate system, or the resulting equations may be more difficult to interpret. Additionally, using rotated coordinate systems may not be necessary if the original coordinate system is already simple and easy to work with.

## 5. Can the process of deriving relations for components by rotation of axes be applied to all physical quantities?

Yes, the process of deriving relations for components by rotation of axes can be applied to any physical quantity that can be represented by vectors, such as displacement, velocity, acceleration, force, or electric field. However, the specific equations used may vary depending on the quantity being analyzed and the specific coordinate system being rotated.

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