# Homework Help: Prove that the distance between point-line is given by some formula

1. Oct 14, 2011

### aero_zeppelin

1. The problem statement, all variables and given/known data

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 dont know how to start proving this...

thanks a lot

2. Oct 14, 2011

### SteamKing

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

3. Oct 14, 2011

### vela

Staff Emeritus
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.

4. Oct 14, 2011

### uart

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

5. Oct 14, 2011

### vela

Staff Emeritus
You mean sin θ, right?

6. Oct 14, 2011

### aero_zeppelin

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

7. Oct 14, 2011

### 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??

8. Oct 14, 2011

### Staff: Mentor

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?

9. Oct 14, 2011

### aero_zeppelin

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?

10. Oct 14, 2011