Final answer:
To find the equivalent rotation matrix when a point is rotated about the x, y, and z-axes by 45 degrees respectively, multiply the basic rotation matrices in the order of the applied rotations.
Step-by-step explanation:
To find the equivalent rotation matrix when a point is rotated about the x, y, and z-axes by 45 degrees respectively, we can use the basic rotation matrices. Let's denote Rx, Ry, and Rz as the rotation matrices for rotation about the x, y, and z-axes respectively. The equivalent rotation matrix can be obtained by multiplying these rotation matrices together in the order of the applied rotations:
Rotation Matrix = Rz(45°) * Ry(45°) * Rx(45°)
The question involves finding an equivalent rotation matrix for a point rotated about the x, y, and z-axes by 45 degrees each. In three-dimensional space, rotation matrices are used to perform rotations around the principal axes (x, y, and z). Each rotation can be represented by a matrix:
Rotation about the x-axis by an angle θ is given by:
Rx(θ) = ⟨ cos(θ) -sin(θ) 0 sin(θ) cos(θ) 0 0 0 1 ⟩
Rotation about the y-axis by an angle θ is given by:
Ry(θ) = ⟨ cos(θ) 0 sin(θ) 0 1 0 -sin(θ) 0 cos(θ) ⟩
Rotation about the z-axis by an angle θ is given by:
Rz(θ) = ⟨ 1 0 0 0 cos(θ) -sin(θ) 0 sin(θ) cos(θ) ⟩
To get the equivalent rotation matrix for the combined rotations, these matrices are multiplied in the order of z-axis rotation, y-axis rotation, and x-axis rotation. Since each rotation is by 45 degrees, θ would be π/4.
The final calculated matrix represents the overall transformation applied to the point after sequential rotations around the x, y, and z-axes by 45 degrees each.