Question:
A vector \( \vec{a} \) has components \( 2p \) and \( 1 \) with respect to a rectangular Cartesian system. This system is rotated through a certain angle about the origin in the positive direction. If with respect to the new system, \( \vec{a} \) has components \( p+1 \) and \( 1 \), then the values of \( p \) are:
A vector \( \vec{a} \) has components \( 2p \) and \( 1 \) with respect to a rectangular Cartesian system. This system is rotated through a certain angle about the origin in the positive direction. If with respect to the new system, \( \vec{a} \) has components \( p+1 \) and \( 1 \), then the values of \( p \) are:
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Always look for invariant physical quantities. Under orthogonal transformations like rotations or translations of axes, lengths of vectors and angles between lines are perfectly preserved.
Updated On: Jun 8, 2026
- \( p=\pm1 \)
- \( p=-1, p=\frac{1}{3} \)
- \( p=1, p=-\frac{1}{3} \)
- \( p=\frac{1}{2}, p=\frac{3}{2} \)
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The Correct Option is C
Solution and Explanation
Concept:
Rotating a coordinate system about the origin changes the individual directional components of a vector, but its absolute length (magnitude) remains completely invariant.
Step 1: Equating the square of the magnitude across both systems.
In the initial system, components are \( (2p, 1) \): \[ |\vec{a}|^2 = (2p)^2 + 1^2 = 4p^2 + 1 \] In the rotated system, components are \( (p+1, 1) \): \[ |\vec{a}|^2 = (p+1)^2 + 1^2 = p^2 + 2p + 1 + 1 = p^2 + 2p + 2 \]
Step 2: Solving the resulting quadratic equation for \( p \).
Equating both length values: \[ 4p^2 + 1 = p^2 + 2p + 2 \] Bringing all terms to the left side: \[ 3p^2 - 2p - 1 = 0 \] Splitting the middle term: \[ 3p^2 - 3p + p - 1 = 0 \implies 3p(p - 1) + 1(p - 1) = 0 \] \[ (3p + 1)(p - 1) = 0 \] This gives two possible values for \( p \): \[ p = 1 \quad \text{or} \quad p = -\frac{1}{3} \]
Step 1: Equating the square of the magnitude across both systems.
In the initial system, components are \( (2p, 1) \): \[ |\vec{a}|^2 = (2p)^2 + 1^2 = 4p^2 + 1 \] In the rotated system, components are \( (p+1, 1) \): \[ |\vec{a}|^2 = (p+1)^2 + 1^2 = p^2 + 2p + 1 + 1 = p^2 + 2p + 2 \]
Step 2: Solving the resulting quadratic equation for \( p \).
Equating both length values: \[ 4p^2 + 1 = p^2 + 2p + 2 \] Bringing all terms to the left side: \[ 3p^2 - 2p - 1 = 0 \] Splitting the middle term: \[ 3p^2 - 3p + p - 1 = 0 \implies 3p(p - 1) + 1(p - 1) = 0 \] \[ (3p + 1)(p - 1) = 0 \] This gives two possible values for \( p \): \[ p = 1 \quad \text{or} \quad p = -\frac{1}{3} \]
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