Particle pierces a sphere (vector)

In summary, to find the angle between OP and OQ, use the dot product formula and solve for θ using the inverse cosine function.
  • #1
prace
102
0
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
 
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  • #2
. 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.
 

Related to Particle pierces a sphere (vector)

1. What is a "particle pierces a sphere (vector)"?

A "particle pierces a sphere (vector)" refers to a scenario in which a point particle (a particle with no size or dimension) travels through or intersects with a spherical object in a specific direction represented by a vector.

2. How is the vector direction of the particle determined?

The vector direction of the particle is determined by its velocity, which is a combination of its speed and direction of motion. This velocity vector is used to calculate the particle's path and determine if it will pierce the sphere.

3. What factors determine if a particle will pierce a sphere?

The main factors that determine if a particle will pierce a sphere are the particle's velocity and the size and location of the sphere. If the particle's velocity is high enough and its path intersects with the sphere, it will pierce through the sphere.

4. How is the point of intersection between the particle and sphere calculated?

The point of intersection between the particle and sphere is calculated using mathematical equations that take into account the particle's velocity, the sphere's radius, and the distance between the particle's starting point and the center of the sphere. These equations can vary depending on the specific scenario and assumptions made.

5. What applications does the concept of "particle pierces a sphere (vector)" have in science?

The concept of "particle pierces a sphere (vector)" has various applications in fields such as physics, engineering, and computer science. It can be used to simulate and analyze the behavior of particles in different scenarios, such as collisions or interactions with objects in a three-dimensional space. It also has practical applications in fields such as fluid dynamics, where particles moving through a spherical object (such as a droplet of water passing through a bubble) can be studied and understood using this concept.

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