# Orthogonal projection onto line L

• kuhle3133
In summary, the conversation discusses the definition of "orthogonal projection onto L" in Euclidean space and how it fails to meet conditions (iii), (i), and (ii) in definitions 1 and 3, making it not a rigid motion. The first condition is that if P and Q are elements of F, then the second coordinate of P is not the second coordinate of Q. The second condition is that if P is an element of E, then there is an element of F having P as its second coordinate. The third condition is that if (P,P') and (Q,Q') are two elements of F, then the segments PQ and P'Q' are congruent.
kuhle3133

Def1. Let L be a line in E. We define the "orthogonal projection onto L" to be
Ol = {(P,Q)| P,Q in E and either
1.P lies on L and P=Q or
2.Q is the foot of the perpendicular to L through P.

Problem 1. Let L be a line in E. Show that Ol is not a rigid motion because it fails condition (iii) of definition 1 and it fails conditions (i) and (ii) of definition 3

Def.1 part (iii) = If P and Q are elements of F, then the second coordinate of P is not the second coordinant of Q.
Def. 3 part (i) = If P is an element of E, then there is an element of F having P as its second coordinant
Def. 3 part (ii) = If (P,P') and (Q,Q') are two elements of F then the segment PQ and P'Q' are congruent.

## The Attempt at a Solution

I can prove that it fails Def 3 part ii because rigid motions preserve distances where orthogonal projections don't, but could use help on the other two.
Thanks

kuhle3133 said:

Def1. Let L be a line in E. We define the "orthogonal projection onto L" to be
Ol = {(P,Q)| P,Q in E and either
1.P lies on L and P=Q or
2.Q is the foot of the perpendicular to L through P.

This is not very well phrased. I think that you mean you are given P and Q is its orthogonal projection onto L.

Problem 1. Let L be a line in E. Show that Ol is not a rigid motion because it fails condition (iii) of definition 1 and it fails conditions (i) and (ii) of definition 3

Def.1 part (iii) = If P and Q are elements of F, then the second coordinate of P is not the second coordinant of Q.[/quote]
You say above that "L is a line in E" which I take to be a Euclidean space, but what is F?
In any case, if P lies on L, the P= Q in which case the second coordinate of P is the second coordinate of Q. Or is there something about F you haven't told us that prohibits this?

Def. 3 part (i) = If P is an element of E, then there is an element of F having P as its second coordinant
Def. 3 part (ii) = If (P,P') and (Q,Q') are two elements of F then the segment PQ and P'Q' are congruent.

## The Attempt at a Solution

I can prove that it fails Def 3 part ii because rigid motions preserve distances where orthogonal projections don't, but could use help on the other two.
Thanks

## What is "Orthogonal projection onto line L"?

"Orthogonal projection onto line L" is a mathematical method for projecting a point or vector onto a line in a specific direction. This projection creates a right angle between the line and the projected point or vector.

## How is "Orthogonal projection onto line L" calculated?

To calculate the orthogonal projection onto line L, you first need to determine the orthogonal basis for the line. This can be done by finding the unit vector in the direction of the line. Then, the projection can be calculated using the dot product between the point or vector and the orthogonal basis.

## What is the purpose of "Orthogonal projection onto line L"?

The purpose of orthogonal projection onto line L is to find the closest point or vector on a line to a given point or vector. It is commonly used in fields such as mathematics, physics, and computer science to solve various problems and equations.

## What is the difference between "Orthogonal projection onto line L" and "Orthogonal projection onto a plane"?

The main difference between orthogonal projection onto line L and orthogonal projection onto a plane is the dimensionality of the objects being projected. Line L is a one-dimensional object, while a plane is a two-dimensional object. The calculations for each type of projection are also slightly different.

## How is "Orthogonal projection onto line L" used in real-world applications?

Orthogonal projection onto line L has various applications in fields such as engineering, computer graphics, and statistics. It is used to solve problems involving lines and vectors, such as finding the shortest distance between a point and a line or determining the direction of a vector onto a specific line.

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