The GATE Grind

GATE 2026 PI – Question 48

Manufacturing Processes II · Computer Integrated Manufacturing · 2 marks · Multiple select

Which ONE OR MORE homogeneous robotic transformations involve a rotation through θ about the Y-axis?

  1. $\begin{pmatrix}\cos\theta&-\sin\theta&0&0\\\sin\theta&\cos\theta&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}$
  2. $\begin{pmatrix}1&0&0&0\\0&\cos\theta&-\sin\theta&0\\0&\sin\theta&\cos\theta&0\\0&0&0&1\end{pmatrix}$
  3. $\begin{pmatrix}\cos\theta&0&\sin\theta&0\\0&1&0&0\\-\sin\theta&0&\cos\theta&0\\0&0&0&1\end{pmatrix}$
  4. $\begin{pmatrix}\cos\theta&0&\sin\theta&3\\0&1&0&4\\-\sin\theta&0&\cos\theta&2\\0&0&0&1\end{pmatrix}$

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Correct answer: (C) $\begin{pmatrix}\cos\theta&0&\sin\theta&0\\0&1&0&0\\-\sin\theta&0&\cos\theta&0\\0&0&0&1\end{pmatrix}$; (D) $\begin{pmatrix}\cos\theta&0&\sin\theta&3\\0&1&0&4\\-\sin\theta&0&\cos\theta&2\\0&0&0&1\end{pmatrix}$

Explanation

A Y-axis rotation preserves the y component and mixes x,z with opposite sine signs. The rotation block is $$R_y=\begin{pmatrix}\cos\theta&0&\sin\theta\\0&1&0\\-\sin\theta&0&\cos\theta\end{pmatrix}.$$ Both C and D contain this block. D additionally translates by (3,4,2), which does not remove its Y rotation. A rotates about Z; B about X. **C,D**.

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