A vector bar{a} has components 1 and 2p with | vedclass

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(C) The magnitude of a vector remains invariant under the rotation of the coordinate system about the origin.Given the components of vector $\bar{a}$

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$p=\frac{-1}{3}$ or $p=1$

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(C) The magnitude of a vector remains invariant under the rotation of the coordinate system about the origin.Given the components of vector $\bar{a}$ in the original system are $(1, 2p)$,its squared magnitude is:$|\bar{a}|^2 = 1^2 + (2p)^2 = 1 + 4p^2$In the new system,the components are $(1, p+1)$,so its squared magnitude is:$|\bar{b}|^2 = 1^2 + (p+1)^2 = 1 + p^2 + 2p + 1 = p^2 + 2p + 2$Since $|\bar{a}|^2 = |\bar{b}|^2$,we have:$1 + 4p^2 = p^2 + 2p + 2$$3p^2 - 2p - 1 = 0$Factoring the quadratic equation:$(3p + 1)(p - 1) = 0$Thus,$p = -\frac{1}{3}$ or $p = 1$.

$A$ vector $\bar{a}$ has components $1$ and $2p$ 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,$\bar{a}$ has components $1$ and $(p+1)$,then:

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