how to convert the small rotation of axes into capital rotation of axes?

Well, I must say, axes don't usually enjoy rotation parties. But if you want to turn a small rotation into a capital rotation for axes, you'll need to give them a crash course in twirling. Start with some axis-sized dance lessons, maybe teach them the tango or the twist. Once they've mastered the basic moves, add some flare - jazz hands, hip thrusts, and don't forget a few spins for good measure. After a rigorous training regime, your small rotation of axes should be ready to strut their stuff as they embrace their new capital rotation status!

To convert a small (lowercase) rotation of axes into a capital (uppercase) rotation of axes, you can follow these steps:

1. Determine the angle of rotation: First, you need to know the angle of rotation for the small rotation. Let's call this angle "θ".

2. Set up the coordinate system: Visualize the original coordinate system, where the small rotation occurred, with the x-axis and y-axis. Also, visualize the rotated coordinate system, with the capital rotation of axes, as the x'-axis and y'-axis.

3. Determine the direction of rotation: Determine whether the small rotation is counterclockwise (anticlockwise) or clockwise. This will help determine the sign convention for the angle of rotation.

4. Apply the rotation matrix: To convert the small rotation into a capital rotation, you can apply a 2D rotation matrix. The rotation matrix depends on the angle of rotation and the direction of rotation. Here is the general form of the rotation matrix:

[x'] [cos(θ) -sin(θ)] [x]
[y'] = [sin(θ) cos(θ)] [y]

Where x and y are the coordinates in the original coordinate system, and x' and y' are the coordinates in the capital rotated coordinate system.

5. Perform the matrix multiplication: Calculate the new x' and y' coordinates by multiplying the rotation matrix by the original x and y coordinates.

x' = cos(θ)*x - sin(θ)*y
y' = sin(θ)*x + cos(θ)*y

6. Determine the new orientation: The resulting x' and y' coordinates represent the new orientation of the rotated axes in the capital coordinate system.

By following these steps, you can convert a small rotation of axes into a capital rotation of axes. Remember to consider the sign convention for the angle of rotation and the direction of rotation to correctly apply the rotation matrix.

To convert a small rotation of axes into a capital rotation of axes, you need to perform three steps: 1) Find the angle and axis of the small rotation, 2) Construct the rotation matrix corresponding to the small rotation, and 3) Combine multiple small rotations to obtain the capital rotation matrix.

Here's a step-by-step guide on how to do it:

1. Find the Angle and Axis of the Small Rotation:
- Given the small rotation, determine the axis about which the rotation occurs and the angle of rotation.
- The axis can be represented by a unit vector (x, y, z) where (x^2 + y^2 + z^2) = 1.
- The angle of rotation can be measured in radians or degrees.

2. Construct the Rotation Matrix for the Small Rotation:
- Use the axis-angle representation to construct the rotation matrix corresponding to the small rotation.
- The rotation matrix for a small rotation around axis (x, y, z) by an angle θ is given by:

R = I + sin(θ) * K + (1 - cos(θ)) * K^2,

where I is the identity matrix, and K is the skew-symmetric matrix:

K = | 0 -z y |
| z 0 -x |
| -y x 0 |

3. Combine Multiple Small Rotations for the Capital Rotation:
- If you have more than one small rotation, find the rotation matrices for each of them.
- To combine multiple small rotations into a capital rotation, simply multiply the rotation matrices corresponding to each small rotation in the desired order.

R_capital = R_n * ... * R_2 * R_1,

where R_capital is the capital rotation matrix and R_n, R_{n-1}, ..., R_1 are the rotation matrices for the individual small rotations.

By following these steps, you can convert a small rotation of axes into a capital rotation of axes.