Intersection points of two lines in two-space

In summary, to show that the given pairs of lines intersect, you can set their position vector equations equal to each other and solve for the values of s and t. This will give you the coordinates of the point of intersection.
  • #1
adrimare
33
0

Homework Statement



Show that the following pairs of lines intersect. Determine the coordinates of the point of intersection.

L1: r= (-3,-1) + t(3,4)

L2: r= (6,2) + s(3,-2)

Homework Equations



?

The Attempt at a Solution



I know that eventually the two lines will reach the same point at which they will intersect, but I'm not really sure how to get to that point. The x and y of L1 has to equal the x and y of L2, right? Do I have to use the direction vectors somehow or guess and check?
 
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  • #2
adrimare said:
. The x and y of L1 has to equal the x and y of L2, right?

Right.:approve:

You can combine those into a single equation very easily. You are given an expression for the position vector of any point an both lines. At the point point of intersection, those position vectors must be the same (otherwise they represent different points). That means, that for some specific values of [itex]s[/itex] and [itex]t[/itex], you must have

[tex](-3,-1)+t(3,4)=(6,2)+s(3,-2)[/itex]

Solve that equation for [itex]s[/itex] and [itex]t[/itex].
 
  • #3
How would you solve for that, though? I have made them into parametric equations and tried to solve so that both x's and y's have the same s and t values. But there are two variables. Without guess and check, how can I solve for these values? The answer is supposed to be (1.5,5) according to the answer section of the book.
 
  • #4
Each component gives you an equation, so you will have two equations and two unknowns. Solving that system of equations is something you should have learned how to do in High School algebra.
 
  • #5
Am I wrong in using parametric equations? So far, I have tried the substitution method because I can't think of another way to do it. For the x-value of the first line,
I got -3 + 3t = 6 + 3s.
I simplified to get t = (9 + 3s)/3, which turns into t = 3 + s. I went to the y-value of the first line. This has to have the same t-value as the x-value, right? I got -1 + 4t = 2- 2s, which turns into 4t = 3 - 2s, which becomes t = (3-2s)/4. The t-values are not the same. Did I do something wrong or am I completely off track here?
 
  • #6
adrimare said:
The t-values are not the same.

They will be the same for a specific value of [itex]s[/itex]...set them equal to each other and solve for that value.
 

FAQ: Intersection points of two lines in two-space

1. What are intersection points of two lines in two-space?

Intersection points of two lines in two-space are the points where two lines intersect or cross each other in a two-dimensional plane. These points represent the solution to a system of equations where the two lines are represented by their respective equations.

2. How do you find the intersection points of two lines in two-space?

To find the intersection points of two lines in two-space, you need to set their equations equal to each other and solve for the variables. This can be done using various methods such as substitution or elimination. The resulting values for the variables will give the coordinates of the intersection point.

3. Can two lines in two-space have more than one intersection point?

Yes, two lines in two-space can have more than one intersection point if they are not parallel or coincident. Depending on the orientation and slope of the lines, they can intersect at different points along their lengths.

4. What does it mean if two lines in two-space have no intersection points?

If two lines in two-space have no intersection points, it means they are parallel and will never intersect. This also means that their equations will have no solution when set equal to each other, as their slopes are equal and they will never cross each other in a two-dimensional plane.

5. Can two lines in two-space intersect at an infinite number of points?

No, two lines in two-space can only intersect at a maximum of one point. If two lines intersect at more than one point, then they are not considered distinct lines and are instead the same line. This is known as being coincident, meaning they overlap each other.

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