The components of a vector vec{a} with respec | vedclass

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(B) The magnitude of a vector remains invariant under the rotation of the coordinate axes.Given the components in the original system are $(2p, 1)$,th

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$p = 1$ or $p = -1/3$

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(B) The magnitude of a vector remains invariant under the rotation of the coordinate axes.Given the components in the original system are $(2p, 1)$,the squared magnitude is $|\vec{a}|^2 = (2p)^2 + 1^2 = 4p^2 + 1$.In the new system,the components are $(p + 1, 1)$,so the squared magnitude is $|\vec{a}|^2 = (p + 1)^2 + 1^2 = p^2 + 2p + 1 + 1 = p^2 + 2p + 2$.Equating the two expressions for the squared magnitude:$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 $p = -1/3$.

The components of a vector $\vec{a}$ with respect to a rectangular Cartesian system are $2p$ and $1$. The system is rotated about the origin through a certain angle in the counter-clockwise direction. If the components of $\vec{a}$ with respect to the new system are $p + 1$ and $1$,then:

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