Question

A vector v=3i 2j 7k is rotated by 60 about the z-axes of the reference frame. it is then rotated by 30 about the x-axes of the reference frame. find the rotation transformation.

Answer 1

To find the rotation transformation for a vector, the respective rotation matrices for rotation about the z-axis and x-axis are obtained and multiplied. This provides a final rotation matrix representing the composite rotation that, when applied to the original vector, gives the rotated vector's new coordinates.

Step-by-step explanation:

To understand the rotation of a vector in three-dimensional space, we use rotation matrices that correspond to rotating the coordinate axes. A vector can be rotated in a sequence of operations where each rotation is described by a matrix that operates on the vector. The rotation transformation for the vector v can be found by first rotating the vector by 60 degrees about the z-axis, followed by a rotation of 30 degrees about the x-axis. To find the overall transformation we can multiply the respective rotation matrices for each rotation.

The rotation matrix Rz(θ) for rotating by an angle θ about the z-axis in a right-handed coordinate system is:

Rz(θ) =
[[cos(θ), -sin(θ), 0],
[sin(θ), cos(θ), 0],
[0, 0, 1]]

The rotation matrix Rx(φ) for rotating by an angle φ about the x-axis is:

Rx(φ) =
[[1, 0, 0],
[0, cos(φ), -sin(φ)],
[0, sin(φ), cos(φ)]]

By substituting θ = 60° and φ = 30° into these matrices and multiplying them, we get the overall rotation matrix that represents the composite rotation transformation. This final matrix, when applied to the original vector, will provide the coordinates of the rotated vector.

Answer 2

To find the rotation transformation for a vector v rotated by 60° around the z-axis and then by 30° around the x-axis, apply the rotation matrices Rz(60°) and Rx(30°) to the vector v.

Step-by-step explanation:

To find the rotation transformation, we can apply the rotation matrices corresponding to the given rotations around the z-axis and x-axis.

For rotating a vector v by an angle θ about the z-axis, the rotation matrix is:

Rz(θ) = [[cos(θ), -sin(θ), 0], [sin(θ), cos(θ), 0], [0, 0, 1]]

For rotating a vector v by an angle θ about the x-axis, the rotation matrix is:

Rx(θ) = [[1, 0, 0], [0, cos(θ), -sin(θ)], [0, sin(θ), cos(θ)]]

To find the final rotated vector, we can apply the rotation matrices in the order mentioned, starting with Rx(30°) and then Rz(60°):

Final rotated vector = Rz(60°) * Rx(30°) * v