Prove that the direction ratios of the line of intersection of two planes $\vec r\cdot(a\hat i+ b\hat j+ c\hat k)=m $ and $\vec r\cdot(d\hat i+ e\hat j+ f\hat k)=n$ is given by $\begin{vmatrix}\hat i&\hat j&\hat k\\a&b&c\\d&e&f\end{vmatrix}$ where m and n are any two real numbers.
Transforming into coordinate form, the vectors are $ax+by+cz=m$ and $dx+ey+fz=n$ .Assuming $x=t$ we can solve for $x$, $y$ and $z$ by converting into coordinate form but that method will be very long. Is there a more intuitive way to do this?
Hint: The line of intersection of two planes is perpendicular to the normal of each of the planes.