Orthogonality, point on line closest to point in space

In summary, the conversation discusses finding the point on a given line that is closest to a given point using vector projection and the concept of orthogonal vectors. However, it is suggested to approach the problem using the slope of the line and finding the equation of a perpendicular line to find the desired point.
  • #1
s_stylie0728
15
0

Homework Statement


Find the point on the line y = 2x+1 that is closest to the point (5,2)


Homework Equations


Vector Projection
(x^Ty/y^Ty)*y
x and y are orthogonal (angle between them 90 degrees) if:
x dot y = 0

The Attempt at a Solution


There's a similar example in my book, but it has information that I'm missing. It gives an additional vector in the direction of the line. Then, in order to find the point on the line that's closest to the point given, he just takes the vector projection of v (which would be (5,2) in my case) onto w (vector in the direction of the line).

This seems relatively straightforward to me, I just don't know how to obtain the value for the vector in the direction of the line. Any guidance? I'd appreciate it!

Thanks!
 
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  • #3
If it were me, I wouldn't worry about "vectors" or "projections". The slope of the given line, y= 2x+ 1, is 2. The slope of any line perpendicular to that is -1/2. What is the equation of a line with slope -1/2 through (5, 2)? Where does that line intersect y= 2x+ 1?
 
  • #4
Oh, wow, duh. I guess I let the language get the best of me. Thank you!
 

1. What is orthogonality and how is it related to points and lines?

Orthogonality refers to the perpendicular relationship between two lines or vectors. In the context of points and lines, it refers to the shortest distance between a point in space and a line. This distance is known as the orthogonal distance or the point-line distance.

2. How is the point on line closest to a point in space determined?

The point on line closest to a point in space is determined by finding the perpendicular distance between the point and the line. This can be done by using the formula for point-line distance, which involves finding the dot product between the vector from the point to a point on the line and the direction vector of the line. The point on the line closest to the point in space will be the point that minimizes the point-line distance.

3. Why is orthogonality important in mathematics and science?

Orthogonality is important in mathematics and science because it allows us to simplify and solve complex problems involving points and lines. It is also a fundamental concept in fields such as linear algebra, geometry, and physics. Many real-world applications, such as computer graphics and data analysis, rely on the concept of orthogonality to find optimal solutions.

4. Can orthogonality be extended to higher dimensions?

Yes, orthogonality can be extended to higher dimensions. In three-dimensional space, orthogonality refers to the perpendicular relationship between planes or vectors. The concept can also be extended to higher dimensions such as four-dimensional space, where it is known as hyperorthogonality.

5. How is orthogonality used in practical applications?

Orthogonality has many practical applications in various fields such as engineering, computer science, and statistics. It is used to solve optimization problems, design efficient algorithms, and analyze data. For example, in computer graphics, orthogonality is used to determine the position and orientation of objects in a 3D scene, while in statistics, it is used to find the best-fit line for a set of data points.

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