Final answer:
To find the composite matrix for rotating 30° about the z-axis and then rotating 60° about the y-axis, multiply the rotation matrices for each rotation together. The composite matrix is: R_y(60°) * R_z(30°) = [[cos(60°)cos(30°), -cos(60°)sin(30°), sin(60°)], [sin(30°)sin(60°)cos(60°)-cos(30°)sin(60°), cos(30°)cos(60°)+sin(30°)sin(60°)sin(60°), -sin(30°)cos(60°)], [-sin(60°)sin(30°)cos(60°)-cos(60°)sin(60°), -cos(60°)sin(30°)+sin(60°)sin(60°)cos(60°), cos(60°)cos(30°)]]
Step-by-step explanation:
To find the composite matrix for rotating 30° about the z-axis and then rotating 60° about the y-axis, we need to multiply the rotation matrices for each rotation together. The general representation of a rotation matrix about the x, y, or z-axis is:
R_x(theta) = [[1, 0, 0], [0, cos(theta), -sin(theta)], [0, sin(theta), cos(theta)]]
R_y(theta) = [[cos(theta), 0, sin(theta)], [0, 1, 0], [-sin(theta), 0, cos(theta)]]
R_z(theta) = [[cos(theta), -sin(theta), 0], [sin(theta), cos(theta), 0], [0, 0, 1]]
To rotate 30° about the z-axis, we use R_z(30°). To rotate 60° about the y-axis, we use R_y(60°). Multiplying these matrices together gives us the composite matrix for the given rotations:
R_y(60°) * R_z(30°) = [[cos(60°)cos(30°), -cos(60°)sin(30°), sin(60°)], [sin(30°)sin(60°)cos(60°)-cos(30°)sin(60°), cos(30°)cos(60°)+sin(30°)sin(60°)sin(60°), -sin(30°)cos(60°)], [-sin(60°)sin(30°)cos(60°)-cos(60°)sin(60°), -cos(60°)sin(30°)+sin(60°)sin(60°)cos(60°), cos(60°)cos(30°)]]