# Distance between a point and a line in space

1. Feb 16, 2010

### lockedup

1. The problem statement, all variables and given/known data
Find the distance between (2,5,1) and the line 2i − 3j + 6k.

2. Relevant equations

3. The attempt at a solutionI can't find a formula to figure this (or one that makes any sense)...

2. Feb 16, 2010

### Gunthi

It's a plane.

3. Feb 16, 2010

### Staff: Mentor

No, it isn't.

4. Feb 16, 2010

### Staff: Mentor

2i - 3j + 6k isn't a line -- it's a vector. It has a certain length, while a line has infinite length. The problem is probably something more like this:
Find the distance between (2,5,1) and the line whose direction is given by the vector 2i − 3j + 6k.​

Just as well. Given that you can't find a formula, how would you approach this problem? According to the forum rules, you have to give it a good shot before anyone can give you any help.

5. Feb 16, 2010

### pootette

Gunthi,

I believe you are asking how to find the distance between a point in space, and a vector?

If so, start by looking at line-distance formulas and vector math.

I hope this gives you a jumping-off point.

6. Feb 16, 2010

### lockedup

My assignment sheet says line...

Does 20 or so google searches count? I've clicked on numerous links, some from here, and none of it makes any sense.

The formula in my Calculus book states:

$$D = \frac{||PQ \times u||}{||u||}$$

P is a point on the line, Q is the point in space, and u is the direction vector. Since I'm only given a vector and as opposed to a line, can I use (0, 0, 0) for P so that PQ is just Q?

7. Feb 16, 2010

### Staff: Mentor

Sure, give your formula a shot.

And no, Google searches don't count...

8. Feb 16, 2010

### Gunthi

You're right, I confused notation, sorry lockedup.

9. Feb 16, 2010

### Gunthi

That was not my question.

10. Feb 16, 2010

### Staff: Mentor

Gunthi,
If it were 2x - 3y + 6z = 0, you would be right

11. Feb 16, 2010

### Gunthi

Yes, that was what I thought initialy.
I'm just not accostumed to working with i,j,k.

12. Feb 16, 2010

### pootette

The formula wants you to multiply the scalar (point) by a unit vector and cross multiply with the given vector. Take the magnitude of the resultant vector. Then divide by magnitude of the unit vector (just a step that has to be done - balances things out . This will give a scalar quantity of distance.

Last edited: Feb 16, 2010