Clockwise Rotation Matrix

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Clockwise Rotation Matrix

In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix.

rotates points in the xy-Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system.

To perform the rotation using a rotation matrix R, the position of each point must be represented by a column vector v, containing the coordinates of the point.

A rotated vector is obtained by using the matrix multiplication Rv. Since matrix multiplication has no effect on the zero vector (i.e., on the coordinates of the origin), rotation matrices can only be used to describe rotations about the origin of the coordinate system.

Rotation matrices provide a simple algebraic description of such rotations, and are used extensively for computations in geometry, physics, and computer graphics.

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In two-dimensional space, a rotation can be simply described by an angle θ of rotation, but it can also be represented by the four entries of a rotation matrix with two rows and two columns.

In three-dimensional space, every rotation can be interpreted as a rotation by a given angle about a single fixed axis of rotation, and hence it can be simply described by an angle and a vector with three entries.

However, it can also be represented by the nine entries of a rotation matrix with three rows and three columns.

The notion of rotation is not commonly used in dimensions higher than 3; there is a notion of a rotational displacement, which can be represented by a matrix, but not associated single axis or angle.

In this matrix conversion of angle 'θ' changes the rotation of Point over plane. For different values of 'θ' different matrices are obtained.

In a clockwise Rotation Matrix, rotation of matrix is considered to be clockwise and angle in clockwise direction is considered as negative for example –900.

Now in above matrix on substituting value of 'θ' in Sin θ that is -900 as it is in clockwise direction since we know that sin (-θ) = - Sin θ.

Therefore we get – Sin 90 that is equals to -1 and finally result obtained for -Sin θ is obtained as 1. Similarly, Cos θ on substituting value of 'θ' as – 90. Since we know that Cos (-θ) gives Cos θ.

Therefore, Cos (-90) is equals to Cos 90 that is equals to 0. Transpose of a matrix is defined as conversion of rows with columns of matrix.

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That is rows become columns of matrix and columns become rows of matrix. For a clockwise rotation matrix, value of 'θ' is 1800 gives similar values.

It is rotation of angle as half turn in either direction. Value of 2700 remains same as 900 with a small difference of signs as it lies below axis while 900 lies above the axis. Thus, result obtained for above matrix is as follows:

Property 2: If we find the sum of element of any row or column then we will get the sum as 1.

Property 3: If we want to find the dot product of any two rows or column then it is always zero.

Property 4: In rotation matrix, rows of rotation matrix stand for coordinate which is in the original space.

Property 5: In rotation matrix, column stands for coordinates which is in rotated space.

We can use rotation matrix in 3D mathematics. If there is a matrix rotation, having three rows then row one will represent ‘right’, row two will represent upward direction and row three will represent the direction towards outside.

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