Perspective view of an object is shown below. The object is successively rotated 180 degrees clockwise about x-axis, 90 degrees clockwise about y-axis and 90 degrees anticlockwise about z-axis. All rotations have to be viewed from a point on the positive axis facing towards the origin. Which one of the following perspective view options will be the result of the rotations?
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To solve complex 3D rotation problems, track the direction of one or two key axes or features of the object through each rotation. Using a physical object like a pen to represent a vector can be very helpful to visualize the transformations.
Step 1: Track a side feature instead of the main pipe.
Let the vector \( \vec{u} = (1, 0, 0) \) represent the direction of a side handle that initially points along the positive x-axis.
Step 2: Apply the 180 degree rotation about the x-axis.
A rotation about the x-axis does not change the x-coordinate, so \( \vec{u} \) stays at \( (1, 0, 0) \).
Step 3: Apply the 90 degree clockwise rotation about the y-axis.
This rotation sends \( (x, y, z) \) to \( (z, y, -x) \), so \( \vec{u} \) becomes \( (0, 0, -1) \).
Step 4: Apply the 90 degree anticlockwise rotation about the z-axis.
This rotation sends \( (x, y, z) \) to \( (-y, x, z) \), so \( \vec{u} \) stays at \( (0, 0, -1) \), since its x and y components are both zero.
Step 5: Read off the final direction.
The handle that started along the positive x-axis now points along the negative z-axis, downward, which matches how the corresponding feature sits in the option where the object has been flipped and turned to lie in that final pose.
\[ \vec{u}_{\text{final}} = (0, 0, -1) \]
This is consistent with the object's final pose shown in Option B.
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Approach Solution -2
A third way to check the rotations is to ask, after all three turns, which of the six directions, up, down, left, right, front or back, the object's original top surface now faces, and compare that to each option's picture.
Option A: shows the original top surface still facing roughly upward, which would only be true if the first 180 degree flip about the x-axis had not been applied at all.
Option B: shows the original top surface now facing toward the viewer's left and slightly back, which is exactly where the top surface ends up after being flipped down by the first rotation, turned to the side by the second, and swung further by the third.
Option C: shows the top surface facing a direction that would only result from swapping the order of the second and third rotations.
Option D: shows the top surface facing forward, which does not account for the sideways turn introduced by the second rotation about the y-axis.
Tracing where the original top surface ends up after all three rotations, in the correct order, points only to Option B.
