I have two planes $\alpha: x-2y+z=2$ and $\beta: 2x+y+z=5$
The angle between the two planes is $80.4°$. The parametric equation for the intersection line between the planes is: $x=-3t, y=1+t, z=4+5t$
Point $P(0,1,4)$ is on the intersection line between the two planes. A sphere is tangent to $\alpha$ in $A(11,9,9)$ and $\beta$ in a point $B$.
$\vec PA$ is perpendicular to both the intersection line and the normal vector of $\alpha$. I need to find the coordinates to the point B.
How can I solve this? My professor said that $\vec PB$ is perpendicular to both the intersection line and the normal vector of $\beta$, but I don't understand how we can know with the given information. Am I missing something?