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

[aero_zeppelin]

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

 

[SteamKing]

Well, you could start by working out what the vectors u and v are.

 

[vela]

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.

 

[uart]

Use u x v = |u| |v| cos(theta)

 

[vela]

Use u x v = |u| |v| cos(theta)

You mean sin θ, right?

 

[aero_zeppelin]

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

 

[aero_zeppelin]

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??

 

[Mark44]

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?

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

any more ideas??

 

[aero_zeppelin]

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?

 

[LCKurtz]

Look at my post #3 in this thread:

https://www.physicsforums.com/showthread.php?t=534771