suppose I have an equation $\frac{\vec{x}\cdot\vec{n_1}}{||\vec{n_2}||}\vec{n_1}=\frac{\vec{x}\cdot\vec{n_2}}{||\vec{n_2}||}\vec{n_2}$ Is this enough to show that $\vec{n_1}=\vec{n_2}$ since $\vec{x}$ is the same in both sides of the equation?
For example is it sufficient to say that since both vectors are equal, and since $\vec{x}$ is the same in both equations then $\vec{n_1}$ must obviously equal $\vec{n_2}$?
On
The two sides of the equation are the orthogonal projections of $\vec x$ onto $\vec n_1$ and $\vec n_2$, respectively, so are scalar multiples of those vectors. In effect, the equation says that $a_1n_1=a_2n_2$ for some scalars $a_1$ and $a_2$ that depend on $\vec x$. Assuming that $\vec n_1$ and $\vec n_2$ aren’t zero, if these scalars are nonzero, then you can conclude that $\vec n_1$ and $\vec n_2$ are parallel, but if they’re zero, the most you can say is that they’re both of these vectors are orthogonal to $\vec x$.
Hint: What if $x = (1,0,0), n_1 = (0,1,0)$ and $n_2 = (0,0,1)$?