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MathematicalPhysicist
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I know how the rotation matrix looks like in the 2x2 and 3x3 orders, but how does it look in general?
thanks in advance.
thanks in advance.
maze said:So that raises an interesting question - what is the minimum number of sequential plane rotations required to represent a general rotation in Rn?
In R2, its just 1. In R3 its 2 since you rotate xhat to xhat', and then roll around xhat'. What about higher dimensions?
A rotation matrix is a mathematical tool used to describe the rotation of an object in three-dimensional space. It is a square matrix that contains a combination of cosine and sine functions, and is used to transform coordinates from one coordinate system to another.
Rotation matrices work by multiplying a vector representing a point in space by the rotation matrix. This results in a new vector that represents the rotated point. The rotation matrix itself is made up of a combination of cosine and sine functions that determine the amount and direction of rotation.
The order of a rotation matrix refers to the size of the matrix (e.g. 2x2, 3x3, etc.). This determines the dimension of the space in which the rotation is taking place. For example, a 2x2 rotation matrix would be used for two-dimensional rotations, while a 3x3 rotation matrix would be used for three-dimensional rotations.
To create a rotation matrix for a specific angle, you can use the formula:
R = [cos(theta) -sin(theta)
sin(theta) cos(theta)]
Where theta is the desired angle of rotation. This will create a 2x2 rotation matrix. For higher order rotation matrices, the process is more complex and may involve using trigonometric identities.
Rotation matrices have many practical applications, including computer graphics, robotics, and physics. They can be used to rotate objects in computer animations, determine the position of a robot arm, or calculate the angular momentum of an object in motion. They are also useful for solving problems in vector calculus and mechanics.