A sphere between two non-paralell planes - finding the centre

[Nikitin]

Allright, I've got mock-exams coming up this winter and I'm struggling with the damn vectors. I made a thread about this before but it turned out terrible due to my poor English skills and because I wrote it in a rush. Anyway:


The problem: Plane α: x-2y+z=2. Plane β: 2x+y+z=5

Point P= (0,1,4) is on the line of intersection, which has the directional vector L, between the two planes.

So L=directional vector of the line of intersection between the two planes.

A sphere is touching plane alpha at point A and plane beta at point B. Show that PB is perpendicular to L and the directional vector of plane Beta, and that PA is perpendicular to L and the directional vector of plane Alpha. Find the coordinates for the centre of the sphere, point S.

point A=(11,9,9) point B= (4,-14,-1).

My work: I found L by finding the cross product between the 2 directional vectors of the planes. I also confirmed that vectors PB & PA are both perpendicular to L, and PB is perpendicular to the directional vector of plane alpha while PA is perpendicular to the directional vector of plane beta.


Where I'm stuck: I set the centre coordinates as: x,y,z. Using some geometry I set up a parametric function where x= 7t y=21t+1 z=4.

I then figured that the radius of the sphere equals= (11-x)^2 + (9-y)^2 + (9-z)^2 = |SA| = |SB| = (-4-x)^2 + (14-y)^2 + (-1-z)^2. This is derived from the formula of a sphere.

Well, I tried to do the algebra, substituting x with 7t, y with 21t+1 and z with 4 but I just ended up with 0t=0.

I tried finding the radius as well, but these attempts too failed.

I spent almost 2 hours on this but I simply couldn't do it, it was (and still is) too hard for me. Can somebody here please help?

 

[mighty2000]

I understand your struggle with vectors and their applications in geometry. It can be a challenging topic, but with practice and understanding of the concepts, I believe you can overcome this difficulty.

From your work, it seems like you have a good understanding of the problem and have made some progress in finding the directional vector L and confirming the perpendicularity of PB and PA to L and the directional vectors of the planes. However, your approach to finding the centre coordinates and radius of the sphere seems to be causing confusion.

Firstly, let's look at the parametric equation you set up for the centre coordinates:

x= 7t
y= 21t+1
z= 4

This is a valid parametric equation, but it represents a line in 3D space, not a point. In order to find the centre coordinates, we need to find the point of intersection between this line and the line of intersection between the two planes. This can be done by setting the equations of the two lines equal to each other and solving for t.

Once you have the value of t, you can substitute it back into the parametric equation to find the coordinates of the centre point, S.

As for finding the radius, you can use the distance formula between point S and either point A or B to find the radius. It is important to note that the distance between a point and a sphere is always equal to the radius of the sphere.

I hope this helps in your understanding and progress with this problem. Good luck on your mock-exams! Remember to take your time and practice as much as possible. You can do it!