Prove that the distance between point-line is given by some formula

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The discussion revolves around proving the formula for the distance between a point P and a line using vectors. It begins with defining vectors u and v, where u connects a point on the line to point P, and v is parallel to the line. Participants clarify that the cross product u x v represents the area of a parallelogram formed by these vectors, and the norm of v is simply its length. There is a focus on relating these concepts to derive the distance formula, emphasizing the need for an algebraic proof rather than just a visual representation. The conversation highlights confusion over geometric interpretations and the relationship between the vectors involved.
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Homework Statement



Show that if u is a vector from any point on a line to a point P not on the line, and v is a vector parallel to the line, then the distance between P and the line is given by

NORM of u x v / NORM v

u x v--> cross product of u and v


I know how to calculate the distance between a point and a line, but I just don't know how to start proving this...

any help please?

thanks a lot
 
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Well, you could start by working out what the vectors u and v are.
 
Do you need to formally prove this or is a picture showing it's true enough? It's pretty easy to show why it's true using a diagram.
 
Use u x v = |u| |v| cos(theta)
 
uart said:
Use u x v = |u| |v| cos(theta)
You mean sin θ, right?
 
vela said:
You mean sin θ, right?


yeah, its sinθ ... but I still don't quite get how to do it...

And also, yes, I believe you have to prove it algebraically or something
 
Ok, so we know that the norm of uxv is the area of the parallelogram formed by u and v...
The norm of v has no geometric interpretation (its just the length of a line)

How can we relate this so it gives the distance between P and the line...

any more ideas??
 
aero_zeppelin said:
Ok, so we know that the norm of uxv is the area of the parallelogram formed by u and v...
The norm of v has no geometric interpretation (its just the length of a line)
No and no. The norm of a vector v has a perfectly good geometric interpretation - it's the length of the vector. A line has infinite length - maybe you meant line segment?
aero_zeppelin said:
How can we relate this so it gives the distance between P and the line...

any more ideas??
 
Mark44 said:
No and no. The norm of a vector v has a perfectly good geometric interpretation - it's the length of the vector. A line has infinite length - maybe you meant line segment?

yeah, line segment. Anyway... I just don't know how to relate both of these.

Is the norm of v considered the "base" of the parallelogram?
 

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