Particle pierces a sphere (vector)

[prace]

A particle starts from point (-2,1,2) with velocity vector v = < 1,1,0 > and hits the plane x + z = 2 and bounces off and penetrates through a sphere x² + y² + z² = 25 at two points, P and Q. What is the angle between OP and OQ.

This was a question on one of my exams that I did not get. I cannot remember completely, but I think the OP and OQ are vectors sharing the origin of the coordinate system as their vertex and we want to find that angle between them.

So I am trying to work this one out and I am getting really stuck.

I started by parametrizing the starting point and vector and came up with:

x = -2 + t
y = 1 + t
z = 2

where t = 2

I also know that the angle of incidence will = the angle of reflectance, so v = -u = < -1,1,0 > (is this statement true?)

But I don't see how I can relate this to the angle between P and Q

 

[lippyka]

. Any help would be really appreciated. Thank you!Yes, the statement that the angle of incidence = the angle of reflectance is true. The angle between OP and OQ can be found using the dot product.Let P and Q be two points on the surface of the sphere: (x1, y1, z1) and (x2, y2, z2). Let OP = (x1 - (-2), y1 - 1, z1 - 2) and OQ = (x2 - (-2), y2 - 1, z2 - 2). Then the angle between OP and OQ is given by the following formula: cos(θ) = (OP · OQ) / (|OP| |OQ|) where θ is the angle between OP and OQ. You can then solve for θ using the inverse cosine function.