Basic vectors question

Question

The components of a vector along $x$ and $y$ directions are $(n+1)$ and $1$ respectively. If the $xy$ coordinate system is rotated by an angle $θ=60°$ then the components change to $n$ and $3$. The value of $n$ is?

Answer 1

I think there is something wrong with your angle $\theta = 60°$.

Nevertheless the problem can be turned into a meaningful one.

Put $u = \begin{pmatrix} n+1 \\ 1\end{pmatrix}$, $v= \begin{pmatrix} n \\ 3\end{pmatrix}$. Since $v$ is obtained from $u$ by rotating, you have $$|u|^2 = |v|^2 \Leftrightarrow (n+1)^2+1 = n^2 + 9 \Leftrightarrow 2n=7 \Leftrightarrow \boxed{n= \frac{7}{2}}$$ But, this corresponds to an angle of $\boxed{\theta \approx 28.1°}$.

Answer 2

You get something like: $\begin{pmatrix}\frac12&\frac{\sqrt3}2\\ -\frac{\sqrt3}2&\frac 12\end{pmatrix}\begin{pmatrix}n+1\\1\end{pmatrix}=\begin{pmatrix}n\\3\end{pmatrix}$

Multiply through and solve for $n$.

So, from the first coordinate, we get $n=1+\sqrt3$. On the other hand, setting the second coordinates equal yields $n=-\frac{5+\sqrt3}{\sqrt3}$.

Apparently we have $2$ equations in $1$ unknown, and it's not consistent. So it's not possible...