I have an equation like $a_ry_1 + b_ry_2 + c_ry_3 = 0$, where $a_r, b_r, c_r$ are constants, I need to find $y_1, y_2, y_3$ where $\hat{y} = y_1 \hat{r} + y_2 \hat{t} + y_3 \hat{n}$ and $\hat{x} = a_r \hat{r} + a_t \hat{t} + a_n \hat{n}$
Earlier I did the following problem,
$$\hat{y}\cdot \hat{x} = 0$$ or, $$(y_1\hat{r} + y_2\hat{t}) \cdot (a_r \hat{r} + a_t \hat{t}) = 0$$ or, $$y_1 a_r + y_2 a_t = 0$$ or, $$y_1 a_r = - y_2 a_t$$ So we can write $$y_1 = - a_t$$ and $$y_2 = a_r$$ where $\hat{x} = a_r \hat{r} + a_t \hat{t}$
can I apply this way to find the current 3D problem (that I explained in the first paragraph) to find the values of $y_1, y_2, y_3$. How should I find those values. I request any suggestions.
Thanks a lot!