The distance between a line and a point

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Homework Help Overview

The discussion revolves around finding the shortest distance between a point and a line in a geometric context, specifically involving vector representations and projections. The problem touches on concepts from coordinate geometry and vector mathematics.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the use of the cross product and dot product in relation to vectors representing the line and the point. There are attempts to clarify the conditions under which the methods apply, particularly regarding the line's position relative to the origin. Questions arise about constructing the normal vector and the implications of perpendicularity in vector calculations.

Discussion Status

The discussion is active, with participants exploring different mathematical approaches and questioning the assumptions underlying their methods. Some guidance has been offered regarding vector forms and relationships, but no consensus has been reached on a specific method or solution.

Contextual Notes

Participants note potential constraints such as the line not passing through the origin and the dimensionality of the vectors involved. There is also mention of the limitations of using the cross product in two-dimensional contexts.

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Homework Statement


So this it the problem, i am not so good at latex, so i decided to use MS office:

dxieqs.png



Homework Equations


the cross product?


The Attempt at a Solution


wr10eq.png

ok i tried this but i got stuck, because i got the bottom right, i am not sure if the top is correct, what if the line (A) does not pass through the origin, then my method won't work, pointing me in the correct direction would be helpfull, also how do you get the projections of V on d, i know how to get the projection of V on A.
 
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I presume that this is the shortest distance from the point [tex](x_{1},y_{1})[/tex] to the line? This happens when you look at the normal to the line passing through the point [tex](x_{1},y_{1})[/tex] which you have noted.

Can you write the line in vector form? If two vectors are perpendicular to each other, what is the value of their dot product? Can you construct the normal from this?

Mat
 
the vector i am trying to find is d, which is perpendicular to A, thus if i do a dot product i would get a zero, but i won't be able to find d.
 
I would instingtively use co-ordainate geometry. The cross product only makes sense for three dimensional vectors.
Write [tex]\mathbf{d}=a\mathbf{i}+b\mathbf{j}[/tex] and then by taking the dot product with the vector representing the line you will be able to find either a or b and use that to find the vector d.
 

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