Question

Consider the following square with the four corners and the center marked as P, Q, R, S and T respectively.
Let X, Y, and Z represent the following operations:
X: rotation of the square by 180 degree with respect to the S-Q axis.
Y: rotation of the square by 180 degree with respect to the P-R axis.
Z: rotation of the square by 90 degree clockwise with respect to the axis perpendicular, going into the screen and passing through the point T.
Consider the following three distinct sequences of operation (which are applied in the left to right order).
(1) XYZ
(2) XY
(3) ZZZZ
Which one of the following statements is correct as per the information provided above?

• A. The sequence of operations (1) and (2) are equivalent
• B. The sequence of operations (1) and (3) are equivalent
• C. The sequence of operations (2) and (3) are equivalent
• D. The sequence of operations (1), (2) and (3) are equivalent

Hint:

When analyzing rotation sequences, consider the total angle of rotation and the axes involved. Sequences that result in the same final orientation are equivalent.

Answer 1

The Correct Option is B

Step 1: Understanding the operations.
- Operation X is a rotation of 180 degrees with respect to the S-Q axis. This operation changes the orientation of the square.
- Operation Y is a rotation of 180 degrees with respect to the P-R axis. This also changes the orientation of the square.
- Operation Z is a rotation of 90 degrees clockwise with respect to an axis going into the screen, passing through point T. This will rotate the square around the specified axis.
Step 2: Analyzing the sequences.
- Sequence (1): XYZ
First, operation X (180 degrees with respect to S-Q) is applied. Then, operation Y (180 degrees with respect to P-R) is applied. Finally, operation Z (90 degrees clockwise with respect to T) is applied. This sequence results in a certain final orientation.
- Sequence (2): XY
This sequence applies operations X and Y only. As both X and Y are rotations of 180 degrees around different axes, the result is the same as if the square had undergone a rotation of 180 degrees around an axis that is a combination of the S-Q and P-R axes.
- Sequence (3): ZZZZ
In this case, four 90-degree rotations are performed around point T, resulting in a full 360-degree rotation, which brings the square back to its original orientation. Therefore, the sequence (3) effectively leaves the square unchanged.
Step 3: Conclusion.
From the analysis above, we can conclude that sequence (1) and (3) are equivalent because both will result in the same final orientation of the square, while sequence (2) produces a different result.