Question:
If a vector \( \vec{v} = 3\hat{i} \) is rotated in the \( x - z \) plane by an angle \( \theta \) with respect to \( x \)-axis in the clockwise direction, then for an observer at \( +y \) axis the vector will be
If a vector \( \vec{v} = 3\hat{i} \) is rotated in the \( x - z \) plane by an angle \( \theta \) with respect to \( x \)-axis in the clockwise direction, then for an observer at \( +y \) axis the vector will be
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To easily visualize rotations in three dimensions, use your right hand: point your thumb along the axis of rotation (the \( +y \) axis in this case). Your curling fingers show the direction of counterclockwise (positive) rotation, which goes from \( +z \) to \( +x \). Therefore, a clockwise (negative) rotation must go from \( +x \) to \( +z \).
Updated On: May 28, 2026
- \( 3\sin\theta\hat{i} \)
- \( 3\cos\theta\hat{i} \)
- \( 3\sin\theta\hat{i} + 3\cos\theta\hat{k} \)
- \( 3\cos\theta\hat{i} + 3\sin\theta\hat{k} \)
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Question:
A vector \( \vec{v} = 3\hat{i} \), initially lying along the positive x-axis, is rotated in the \( x-z \) plane through an angle \( \theta \) in the clockwise direction as seen by an observer looking down from the positive y-axis. We need to find the expression for the rotated vector.
Step 2: Key Formula or Approach:
For an observer situated at the positive y-axis looking down at the \( x-z \) plane:
- The positive x-axis (\( \hat{i} \)) points to the right.
- By the right-hand rule (\( \hat{i} \times \hat{j} = \hat{k} \)), the positive z-axis (\( \hat{k} \)) points out of the page towards the observer, which corresponds to the "down" direction on the floor.
- A clockwise rotation of the vector \( \vec{v} = 3\hat{i} \) (which initially points to the right) by an angle \( \theta \) rotates it towards the positive z-axis (downwards on the floor).
Step 3: Detailed Explanation:
Since the vector of magnitude 3 is rotated clockwise from the positive x-axis towards the positive z-axis:
- The component along the positive x-axis becomes:
\[ v_x = 3 \cos\theta \]
- The component along the positive z-axis becomes:
\[ v_z = 3 \sin\theta \]
Thus, the rotated vector \( \vec{v}' \) is given by:
\[ \vec{v}' = v_x \hat{i} + v_z \hat{k} = 3\cos\theta\hat{i} + 3\sin\theta\hat{k} \]
Step 4: Final Answer:
The rotated vector is \( 3\cos\theta\hat{i} + 3\sin\theta\hat{k} \), which corresponds to option (D).
A vector \( \vec{v} = 3\hat{i} \), initially lying along the positive x-axis, is rotated in the \( x-z \) plane through an angle \( \theta \) in the clockwise direction as seen by an observer looking down from the positive y-axis. We need to find the expression for the rotated vector.
Step 2: Key Formula or Approach:
For an observer situated at the positive y-axis looking down at the \( x-z \) plane:
- The positive x-axis (\( \hat{i} \)) points to the right.
- By the right-hand rule (\( \hat{i} \times \hat{j} = \hat{k} \)), the positive z-axis (\( \hat{k} \)) points out of the page towards the observer, which corresponds to the "down" direction on the floor.
- A clockwise rotation of the vector \( \vec{v} = 3\hat{i} \) (which initially points to the right) by an angle \( \theta \) rotates it towards the positive z-axis (downwards on the floor).
Step 3: Detailed Explanation:
Since the vector of magnitude 3 is rotated clockwise from the positive x-axis towards the positive z-axis:
- The component along the positive x-axis becomes:
\[ v_x = 3 \cos\theta \]
- The component along the positive z-axis becomes:
\[ v_z = 3 \sin\theta \]
Thus, the rotated vector \( \vec{v}' \) is given by:
\[ \vec{v}' = v_x \hat{i} + v_z \hat{k} = 3\cos\theta\hat{i} + 3\sin\theta\hat{k} \]
Step 4: Final Answer:
The rotated vector is \( 3\cos\theta\hat{i} + 3\sin\theta\hat{k} \), which corresponds to option (D).
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