Intersecting & Finding Parallel Lines in R^3 Space

In summary: If so, then it is parallel to the vector going from (0, 0, 0) to (9, 6, 0). If not, then it is not parallel.
  • #1
jonney
3
0
given two lines x=s(1,2,1) and x=(960)+t(0,1,-1)
how would i go about finding a vector that intersects both lines and then finding another line parallel to that vector? help would be much appreciated
 
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  • #2
You cannot add 960 + t(0,1,-1) since you cannot add a scalar and a vector meaningfully.
 
  • #3
I have a feeling that that is supposed to be (9, 6, 0). In any case, to find a vector that intersects both lines, just pick a point on each line, and then find the line connecting the two.
 
  • #4
Moderator's note:

Before offering more help, let's let the OP respond with an attempt at solving the problem.
 
  • #5
here is what i have come up with is this right

general equation for line 1 x-y+z=0, parametric equation x=s y=2s z=s
general equation for line 2 x+y+z=15, parametric equation x=9 y=6+t z=-t

then created a line with the vector (1,-1,1) going through the origin.
x =(0,0,0) +v(1,-1,1)
therefor general equation is x+2y+z=0, parametric x=v y=-v z=v

crossing of line 1 and line 3
s+4s+s=0
6s=0
s=0
therefore x=0 y=0 z=0

crossing of line 2 and line 3
v-v+v=15
v=15
therefore x=15 y=-15 z=15

so the third line crosses line one at (0,0,0) and line 2 and (15,-15,15)
is this correct?
appreciate all the help
 
  • #6
jonney said:
here is what i have come up with is this right

general equation for line 1 x-y+z=0, parametric equation x=s y=2s z=s
general equation for line 2 x+y+z=15, parametric equation x=9 y=6+t z=-t
Your general equations above are equations of planes, not lines. Your parametric equations represent the lines, though.

BTW, in your original post you said
jonney said:
... x=(960)+t(0,1,-1)
Without commas, the 960 part was confusing to several posters.


jonney said:
then created a line with the vector (1,-1,1) going through the origin.
x =(0,0,0) +v(1,-1,1)
therefor general equation is x+2y+z=0, parametric x=v y=-v z=v

crossing of line 1 and line 3
s+4s+s=0
6s=0
s=0
therefore x=0 y=0 z=0

crossing of line 2 and line 3
v-v+v=15
v=15
therefore x=15 y=-15 z=15

so the third line crosses line one at (0,0,0) and line 2 and (15,-15,15)
is this correct?
appreciate all the help

All you need to do is find one point on each line, and then construct a vector from one point to the other. By inspection, (0, 0, 0) is a point on line 1, and (9, 6, 0) is a point on line 2.

Now, form a vector from one point to the other, and then find the equation of the line with this direction that goes through, say, (0, 0, 0). That's what I would do.
 
  • #7
Hey thanks heaps for that, just one more question how do i know that this line is parallel to the vector.
I got x = (0,0,0)+t(9,6,0) for the vector equation of the line does this look right.
also sorry about the confusion on the (960) when it should have been (9,6,0).
 
  • #8
You can check this for yourself. Does your line intersect the two given lines (the ones in your first post)?
 

FAQ: Intersecting & Finding Parallel Lines in R^3 Space

How do I determine if two lines in R^3 space are parallel?

To determine if two lines in R^3 space are parallel, you can use the cross product of their direction vectors. If the cross product is equal to the zero vector, then the lines are parallel. Another way is to check if the ratios of the coefficients of the equations for the lines are equal.

Is it possible for two lines in R^3 space to intersect?

Yes, it is possible for two lines in R^3 space to intersect. This can happen when the lines are not parallel and their equations have a unique solution, meaning there is one point where both lines pass through.

Can two lines in R^3 space be parallel in one direction and intersect in another direction?

No, two lines in R^3 space cannot be parallel in one direction and intersect in another direction. If two lines are parallel in one direction, they will remain parallel in any other direction.

How do I find the point of intersection for two intersecting lines in R^3 space?

To find the point of intersection for two intersecting lines in R^3 space, you can solve the system of equations formed by their equations. The solution will give you the coordinates of the point of intersection.

Can three lines in R^3 space intersect at the same point?

Yes, it is possible for three lines in R^3 space to intersect at the same point. This can happen when the three lines are not parallel and their equations have a unique solution, meaning there is one point where all three lines pass through.

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