(C) The magnitude of a vector remains invariant under the rotation of the coordinate system about the origin. Therefore,the magnitude of $\vec{a}$ bef

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(C) The magnitude of a vector remains invariant under the rotation of the coordinate system about the origin. Therefore,the magnitude of $\vec{a}$ before rotation is equal to the magnitude of $\vec{a}$ after rotation. $|\vec{a}|^2 = (2p)^2 + 1^2 = (p+1)^2 + 1^2$ $4p^2 + 1 = p^2 + 2p + 1 + 1$ $4p^2 + 1 = p^2 + 2p + 2$ $3p^2 - 2p - 1 = 0$ Factoring the quadratic equation: $3p^2 - 3p + p - 1 = 0$ $3p(p-1) + 1(p-1) = 0$ $(3p+1)(p-1) = 0$ Thus,$p=1$ or $p=-\frac{1}{3}$.

Class 12 Mathematics Vector Algebra Basic , Modulus and Algebra of vectors

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$A$ 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 counter-clockwise sense. If,with respect to the new system,$\vec{a}$ has components $p+1$ and $1$,then:

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A
$p=0$
B
$p=-1$ or $p=\frac{1}{3}$
C
$p=1$ or $p=-\frac{1}{3}$
D
$p=1$ or $p=-1$

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$A$ 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 counter-clockwise sense. If,with respect to the new system,$\vec{a}$ has components $p+1$ and $1$,then:

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