If a dice has 3 on the top, right back 6, right front 2, bottom 4, left front 1, left back 5 and is rotated by +180 about the z axis and +90 about the x axis, how many dots does each face have after the rotation?
You talk about front, bottom, left
then you mention x axis, z axis.
Where are the axis?
the z axis is top/ bottom, the x/y axis is right/ left
180° about z takes (x,y,z) -> (-x,-y,z)
90° about x takes (x,y,z) -> (x,z,-y)
so, your two rotations take (x,y,z) -> (-x,z,y)
So, figure where the spots start out, and apply the rotations.
To determine the number of dots on each face of the dice after the given rotation, we need to understand the orientation of the dice and how it changes with each rotation.
Let's break down the rotations one by one:
1. Rotation about the z-axis (+180 degrees):
This rotation flips the dice upside down. The side that was previously facing up (with 3 dots) will now be facing down. Similarly, the side that had 4 dots on the bottom will now be facing up. The other sides remain unchanged.
After this rotation, the new configuration is:
Top: 4 dots
Right: 6 dots
Front: 2 dots
Bottom: 3 dots
Left: 5 dots
Back: 1 dot
2. Rotation about the x-axis (+90 degrees):
This rotation rotates the dice counterclockwise when viewing it from the front. The side that was previously facing up will now be facing left, and the side that was facing right will now be facing up. The other sides will also change accordingly.
After this rotation, the new configuration is:
Top: 1 dot
Right: 4 dots
Front: 2 dots
Bottom: 6 dots
Left: 3 dots
Back: 5 dots
Therefore, after the given rotations, the number of dots on each face of the dice are:
Top: 1 dot
Right: 4 dots
Front: 2 dots
Bottom: 6 dots
Left: 3 dots
Back: 5 dots