distance between parallel lines

**hmmmm** · May 2nd 2010, 06:09 AM

i came across a question today about the distannce between two parallel lines in 2D and i was wondering

if i am asked the distance between two parallel lines, could i find the distance between the two parallel planes that these vectors were on in 3D (if the z component was equal to 0 for both planes)?

**Plato** · May 2nd 2010, 06:23 AM

Originally Posted by hmmmm

i came across a question today about the distannce between two parallel lines in 2D and i was wondering if i am asked the distance between two parallel lines, could i find the distance between the two parallel planes that these vectors were on in 3D (if the z component was equal to 0 for both planes)?

There are well-known formulae for the distance between a point and a line and between a point and a plane. Finding distance between a point on one line to the other line gives the distance between the two parallel lines. The same basic idea works for finding the distance between two parallel planes.,

**HallsofIvy** · May 2nd 2010, 09:05 AM

Originally Posted by hmmmm

i came across a question today about the distannce between two parallel lines in 2D and i was wondering

if i am asked the distance between two parallel lines, could i find the distance between the two parallel planes that these vectors were on in 3D (if the z component was equal to 0 for both planes)?

Two parallel lines are on an infinite number of parallel planes- however, they all have the same distance between them.

Personally, what I would do is choose an arbitrary point on one of the lines, construct the plane perpendicular to the line through that point (very simple since you are given the line and so a vector perpendicular to the plane), determine where the other line intersected the line, then find the distance between those two points.

**hmmmm** · May 2nd 2010, 09:46 AM

well if i had for example two lines 3x+7y=-4 and 3x+7y=10 and treated them as parallel planes, with equation x.n=-4 and x.n=10 can i then use the distance between these two planes, e.g. |-4-10|/|n| where n = <3,4>

**Plato** · May 2nd 2010, 10:22 AM

The point $(1,-1)$ is on the line $3x+7y+4=0$ .
The distance from $(1,-1)$ to $3x+7y-10=0$ is $\frac{|3(1)+7(-1)-10|}{\sqrt{3^2+7^2}}$ .
That is the distance between the two parallel lines.

**hmmmm** · May 2nd 2010, 10:46 AM

ok so that is equal to 14/((58)^(1/2)) but if i use x.n=-4 and x.n=10 where n = <3,4,0> then i get |-4-10|/((9+49)^(1/2)) which is equal to the same thing,

so i was wondering if treated parallel lines as planes parallel to the z-plane if i can use this to find the distance?

**Plato** · May 2nd 2010, 11:43 AM

Originally Posted by hmmmm

ok so that is equal to 14/((58)^(1/2)) but if i use x.n=-4 and x.n=10 where n = <3,4,0> then i get |-4-10|/((9+49)^(1/2)) which is equal to the same thing,
so i was wondering if treated parallel lines as planes parallel to the z-plane if i can use this to find the distance?

Suppose that $R=<x,y,z>$ and $N$ is a vector.
Then $N\cdot(R-Q)=0$ is a plane with normal $N$ containing the point $Q$ .
For any point $P$ its distance to the plane is $\frac{{\left| {\overrightarrow {PQ} \cdot N} \right|}}{{\left\| N \right\|}}$
Just do in general. Don’t worry where the z-plane is.

**hmmmm** · May 2nd 2010, 12:19 PM

sorry what i was tying to ask is can i use the formula for the distance between parallel planes in 3D x.n=c and x.n=d distance = |c-d|/|n| for lines if i treat z=0

**Plato** · May 2nd 2010, 12:30 PM

Originally Posted by hmmmm

sorry what i was tying to ask is can i use the formula for the distance between parallel planes in 3D x.n=c and x.n=d distance = |c-d|/|n| for lines if i treat z=0

That is exactly what I gave you.
Both planes have normal $N$ ; there are parallel.
So take $P$ on one plane and $Q$ on the other plane an use $\frac{{\left| {\overrightarrow {PQ} \cdot N} \right|}}<br /> {{\left\| N \right\|}}$ .
That is the distance between the planes.

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