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?
Show Hint
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: Understanding the Concept:
This problem requires performing a sequence of 3D rotations on an object and determining its final orientation. It's crucial to apply the rotations in the correct order and direction around the specified axes.
Step 2: Detailed Explanation:
Let's track the orientation of a primary feature, such as the main central pipe which is initially vertical (along the +z axis).
Initial Position: The main pipe is aligned with the z-axis. Let's represent its direction as a vector V = (0, 0, 1).
Rotation 1: 180 degrees clockwise about x-axis.
A 180-degree rotation around the x-axis flips the sign of the y and z coordinates.
The new vector V' becomes (0, 0, -1). The main pipe is now pointing straight down.
Rotation 2: 90 degrees clockwise about y-axis.
A clockwise rotation around the y-axis (viewed from +y) transforms a point (x, y, z) to (z, y, -x).
Applying this to V' = (0, 0, -1), the new vector V'' becomes (-1, 0, 0). The main pipe is now horizontal, pointing along the negative x-axis.
Rotation 3: 90 degrees anti-clockwise about z-axis.
An anti-clockwise rotation around the z-axis (viewed from +z) transforms a point (x, y, z) to (-y, x, z).
Applying this to V'' = (-1, 0, 0), the final vector V''' becomes (0, -1, 0). The main pipe is now horizontal, pointing along the negative y-axis.
Analyzing the Final Orientation:
The main pipe, which was originally vertical, is now pointing horizontally away from the default viewing position (along the -y direction).
Let's track another feature, the lower pipe assembly initially pointing right (along the +y axis). Let's call its vector L = (0, 1, 0).
After Rot 1: L' = (0, -1, 0)
After Rot 2: L'' = (0, -1, 0) (rotation is about its own axis)
After Rot 3: L''' = (1, 0, 0). This pipe now points forward along the +x axis.
Matching with Options:
We are looking for a view where the main central pipe points away from us (-y) and the lower assembly has a prominent pipe pointing to the right (+x). Option (B) is the only view that matches this final configuration.
Step 3: Final Answer:
After applying the three successive rotations, the object's final view is depicted in option (B).
