Determine if two lines are parallel, intersecting, or skewed.

The question only specified lines. In summary, the two lines r1=<1+2s,3+2s,2-2s> and r2=<2+t,6-t,-2+t> are not parallel and therefore are skewed and intersecting. However, no point of intersection can be found because no values for s and t exist that satisfy all three equations.
  • #1
getty102
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Homework Statement


Determine if the lines r1=<1+2s,3+2s,2-2s> and r2=<2+t,6-t,-2+t> are parallel, intersecting, or skew. If they intersect, find the point of intersection.


Homework Equations


x: 1+2s=2+t
y: 3+2s=6-t
z: 2-2s=-2+t


The Attempt at a Solution


I set the components of r1=r2 then solved for s and t using substitution.
s=1
t=2
I then used 1+2(1)=2+(2) to get 3≠4
therefore it is skewed? or intersecting?

I'm not sure how to find the point of intersection if it does intersect.
 
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  • #2
The initial process for finding intersection is good (although you didn't tell us which equations you initially used to get candidates for s & t, so checking was impossible). But in essence we need the same two values to satisfy all three equations.

If there is no solution as here, you then need to get direction vectors for the two lines, which consists of the multipliers of the variable parameter here. Then if those vectors are a simple multiple of one another, the lines are parallel, otherwise skewed.

There's also the possibility that the lines are coincident, which will come out of your check for intersection as a relationship between the two parameters (choose any s and you can find a suitable t to satisfy the equations).
 
  • #3
I determined that the two planes stated above are not parallel because the planes are not multiples of each other, therefore they are skewed and intersecting. I'm not what the next step is to find the intersecting line.
 
  • #4
The lines you specify are not intersecting, because you couldn't find values for t and s that made all three equations true. (Or, more accurately, you showed that no such s & t values exist). Finding a unique s & t would have also given you the intersection point.

I don't know about any planes.
 

1. How do you determine if two lines are parallel?

To determine if two lines are parallel, you need to compare their slopes. If the slopes are equal, then the lines are parallel.

2. What is the formula for finding the slope of a line?

The formula for finding the slope of a line is (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.

3. Can two intersecting lines have the same slope?

No, two intersecting lines cannot have the same slope. If they did, then they would be parallel instead of intersecting.

4. How do you determine if two lines are intersecting?

To determine if two lines are intersecting, you need to compare their slopes. If the slopes are different, then the lines will eventually intersect at a single point.

5. What is the difference between parallel and skewed lines?

Parallel lines have the same slope and never intersect, while skewed lines have different slopes and never intersect. Skewed lines are also not parallel because they are not in the same plane.

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