Question:
If a vector \(3\hat{i}+4\hat{j}-5\hat{k}\) is rotated through a certain angle about the origin in the anti-clockwise direction, then the components of the new vector are \(a+1,-3,5\) . The possible values of \(a\) is
If a vector \(3\hat{i}+4\hat{j}-5\hat{k}\) is rotated through a certain angle about the origin in the anti-clockwise direction, then the components of the new vector are \(a+1,-3,5\) . The possible values of \(a\) is
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A rotation keeps the length of the vector unchanged.
Updated On: Oct 1, 2026
- 5 or 3
- 5 or -3
- 4 or -2
- -5 or 3
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept:
Rotating a vector about the origin changes its direction but not its magnitude.
Step 2: Original magnitude squared:
\[ |3\hat i + 4\hat j - 5\hat k|^2 = 9 + 16 + 25 = 50 \]
Step 3: New vector:
The new vector is \((a+1)\hat i - 3\hat j + 5\hat k\), with magnitude squared \((a+1)^2 + 9 + 25\).
\[ (a+1)^2 + 34 = 50 \Rightarrow (a+1)^2 = 16 \Rightarrow a + 1 = \pm4 \]
Step 4: Values of a:
\(a = 3\) or \(a = -5\). This is option (D). Check: \(a = 3\) gives \((4, -3, 5)\) and \(a = -5\) gives \((-4, -3, 5)\), both of squared length \(16 + 9 + 25 = 50\).
Final Answer:
Length is preserved, so (a + 1)^2 = 16. \[ \boxed{\text{(D) }-5\ \text{or}\ 3} \]
Rotating a vector about the origin changes its direction but not its magnitude.
Step 2: Original magnitude squared:
\[ |3\hat i + 4\hat j - 5\hat k|^2 = 9 + 16 + 25 = 50 \]
Step 3: New vector:
The new vector is \((a+1)\hat i - 3\hat j + 5\hat k\), with magnitude squared \((a+1)^2 + 9 + 25\).
\[ (a+1)^2 + 34 = 50 \Rightarrow (a+1)^2 = 16 \Rightarrow a + 1 = \pm4 \]
Step 4: Values of a:
\(a = 3\) or \(a = -5\). This is option (D). Check: \(a = 3\) gives \((4, -3, 5)\) and \(a = -5\) gives \((-4, -3, 5)\), both of squared length \(16 + 9 + 25 = 50\).
Final Answer:
Length is preserved, so (a + 1)^2 = 16. \[ \boxed{\text{(D) }-5\ \text{or}\ 3} \]
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