Distance from a point to a line in r^3

**Kakariki** · June 6th 2010, 08:10 AM

Hey guys. I am curious if I am doing this correctly. For some reason I am doubting my math here.

Question
Find the distance from the point (-3,0,1) to the line $\frac{x-2}{2} = \frac{y+2}{-2} = \frac{z}{-1}$

Solution
I know how to solve the problem when the line is in the form Ax + By + Cz + D, but I am unsure if I put it in that form correctly.

I first took the x and y and put both them equal to zero. So:
$-x - y - 3 = 0$

**Now, this is where I am unsure of myself. Can I set $-x - y - 3$ equal to $\frac {z}{-1}$ ?

If so, I get the equation: $x + y -z + 3 = 0$ . Which I can run through the formula I have, and I get the distance of:
$\frac{-1\sqrt{3}}{3}$

Hopefully someone can clear this up for me! There is no example similar to this in my text. There is with something equal to something else, but not equal to two other things... if you know what I mean. Hard to explain.

Thank you for reading!

**Plato** · June 6th 2010, 08:59 AM

Look at my reply to your other posting.

Here $P(-3,0,1),~Q(2,-2,0)~\&~d=2i-2j-k$

**Kakariki** · June 6th 2010, 09:23 AM

Originally Posted by Plato

Look at my reply to your other posting.

Here $P(-3,0,1),~Q(2,-2,0)~\&~d=2i-2j-k$

The answer I am looking for is whether I am doing the arithmetic to get to the form Ax + By + Cz + D correctly. Thank you for helping though!

**Plato** · June 6th 2010, 09:55 AM

Originally Posted by Kakariki

I know how to solve the problem when the line is in the form Ax + By + Cz + D, but I am unsure if I put it in that form correctly.

I deliberately did not address that statement.
Why? Because it is totally wrong. A line never has that form.
There are three forms for a line.

Vector $(a,b,c)+t(d_1,d_2,d_3)$ .

Symmetric $\frac{x-a}{d_1}=\frac{y-b}{d_2} =\frac{z-c}{d_3};~~d_i\ne0$

Parametric $\ell (t) = \left\{ \begin{gathered}<br /> x = a + td_1 \hfill \\<br /> y = b + td_2 \hfill \\<br /> z = c + td_3 \hfill \\ <br /> \end{gathered} \right.$

Thread: Distance from a point to a line in r^3

LinkBack

Thread Tools

Display

Distance from a point to a line in r^3

Similar Math Help Forum Discussions

Curve's point A tangent line passes through a point P. What is the distance of AP

Distance between point and line

Distance from a point to a line

distance from point to line

Calculating coords of a point on a straight line at a distance from start point...

Search Tags