A vector $\bar{a}$ has components $2 p$ and 1 with respect to a rectangular Cartesian system. This system is…
A vector $\bar{a}$ has components $2 p$ and 1 with respect to a rectangular Cartesian system. This system is rotated though a certain angle about the origin in the counter clockwise sense. If, with respect to the new system, à has components $p+1$ and 1 , then
$p=0$
$\mathrm{p}=-1$ or $\mathrm{p}=\frac{1}{3}$
$\mathrm{p}=1$ or $\mathrm{p}=-\frac{1}{3}$
$\mathrm{p}=1$ or $\mathrm{p}=-1$
Solution
Magnitude of $\vec{a}$ before rotation = Magnitude of $\vec{a}$ after rotation
$\begin{aligned} & \Rightarrow(2 p)^2+1^2=(p+1)^2+1^2 \\ & \Rightarrow 4 p^2+1=p^2+2 p+1+1 \\ & \Rightarrow 3 p^2-2 p-1=0 \\ & \Rightarrow p=1 \text { or } p=\frac{-1}{3}\end{aligned}$