Question

A vector a has components $2\mathrm{p}$ and 1 w.r.t in a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter-clockwise sense. If w.r.t the new system, a has components $\mathrm{p}+1$ and $1$, then

Answer 1

Step 1: Given and assume

Let,

The unit vectors along the coordinate axes are i and j.

$\therefore \overrightarrow{\mathrm{a}}=2\mathrm{pi}+1.\mathrm{j}...\left(1\right)$

On rotation, the vector has components $\mathrm{p}+1$ and $1$is $\mathrm{b}$

$\therefore \overrightarrow{\mathrm{b}}=(\mathrm{p}+1)\mathrm{i}+1.\mathrm{j}...\left(2\right)$

Step 2: Calculation

The magnitude of the initial and end vectors will stay constant during rotation.

$$\begin{gathered} \Rightarrow \left|\stackrel{\rightharpoonup }{\mathrm{b}}\right|=\left|\stackrel{\rightharpoonup }{\mathrm{a}}\right| \\ {\left|\mathrm{b}\right|}^2={\left|\mathrm{a}\right|}^2 \end{gathered}$$

$$\begin{gathered} \Rightarrow {(\mathrm{p}+1)}^2+1={\left(2\mathrm{p}\right)}^2+1 \\ \Rightarrow 3{\mathrm{p}}^2-2\mathrm{p}-1=0 \\ \Rightarrow (3\mathrm{p}+1)(\mathrm{p}-1)=0 \\ \Rightarrow \mathrm{p}=1 \\ \mathrm{p}=-\frac{1}{3}. \end{gathered}$$