# Finding Point P on Plane S: A Reflective Approach

• prace
In summary, the two points A and B are the same distance from plane S, but they have different angles.

## Homework Statement

Given a plane S: x+y+z=8 and two points A(6,5,3) and B(10,5,-1), find the point P on plane S such that a line from point A bounces off of the plane S at point P and passes through point B.

## The Attempt at a Solution

If I know that the angle of incidence = the angle of reflectance, then I can also say that a normal vector from plane S bisects will these two points at point P. So if I can proove that the angle between vectors AP and n is the same as the angle between vectors BP and n then this should solve my problem. Well, this is what I tried to show at least, and did not come up with the angle between the two vectors to be the same.

I found my normal vector n to be <1,1,1> and I took my point P to be (8,0,0). So, from there, I found vectors AP and BP to be <2,-5,-3> and <-2,-5,1> respectively.

Any help on which direction to go towards next would be a great help! Thank you!

Points A and B are in fact the same distance from plane S. (Can you show this?)

Draw a diagram reflecting this, and add point P to the diagram. What do you realize?

I worked out the distances to be different than each other. To find the distance of points A and B from plane S, i first found a point on plane S to reference from, call it Q. So, to find the distance of point A from plane S, i found:

h1 = |QA - proj (QA on n)|, where n is the normal vector for plane S.

I came up with a value of ~32.4.

to find the distance of point B from plane S, i found:

h2 = |QB - proj (QB on n)|, where again, n is the normal vector for plane S.

I came up with a value of ~134.28.

What happened? If I don't have the same height, how can I relate the two angles to each other?

Shouldn't it be $$h_{1} =\ \mid proj\ (\ QA\ on\ n\ ) \mid$$?

Try drawing a diagram, it should help.

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## 1. What is the purpose of finding point P on plane S?

The purpose of finding point P on plane S is to determine the location of a specific point on a two-dimensional surface. This can be useful in various applications such as navigation, geometry, and computer graphics.

## 2. What information is needed to find point P on plane S?

To find point P on plane S, you will need the coordinates of at least three points that lie on the plane. This allows you to create a system of equations and solve for the coordinates of point P.

## 3. Can point P be located using only two points on plane S?

No, at least three points are needed to uniquely determine the location of point P on plane S. With only two points, there are infinite possible locations for point P.

## 4. What is the reflective approach to finding point P on plane S?

The reflective approach involves using the reflection property of a point on a plane to find its location. This means that the distance from point P to any point on the plane is equal to the distance from the reflected point of P to the same point on the plane. By using this property, a system of equations can be created and solved to find the coordinates of point P.

## 5. What are some real-world applications of finding point P on plane S?

Some real-world applications include determining the location of a target in a 2D space for a missile or projectile, finding the intersection point of two roads on a map, and locating specific points on a computer screen for graphic design or gaming purposes.