Finding the intersection points of the two lines in space

[macaulay]

given the lines in space

L1 : x = 2t + 1, y = 3t + 2, z = 4t + 3
L2 : x = s + 2, y = 2s + 4, z = -4s – 1
Find the point of intersection of L1 and L2.
How do i solve this?

 

[member 392791]

set the x's equal to each other and solve for either t or s, then plug into the other variables to get the coordinates

 

[paulfr]

Another method ....
Set the x's and y's equal to each other
2t+1 = s+2
3t+2 = 2s+4
solve the system for t and s
Plug the resulting values for t and s into
z=4t+3 and z = -4s-1
to be sure they give the same value for z
If they do not, the lines are skew
If they do, then the values for t and s give the
coordinates of the intersection point for x,y and z

 

[macaulay]

thank u very much mr. paulfr and woopydalan...i appreciate your replies to my question.. thank u very much =)

 

[CellCoree]

To find the point of intersection of two lines in space, we need to solve the system of equations formed by the parametric equations of the two lines. In this case, the system of equations would be:

2t + 1 = s + 2
3t + 2 = 2s + 4
4t + 3 = -4s - 1

We can solve this system by using elimination or substitution method. Once we solve for the values of t and s, we can substitute them back into the equations of the lines to find the coordinates of the point of intersection.

Alternatively, we can also use vector equations to solve for the intersection point. We can represent each line as a vector with its direction and starting point, and then set the two vectors equal to each other. This will give us a vector equation that we can solve for the values of t and s, and then find the coordinates of the point of intersection.

In either case, it is important to check for any inconsistencies or no solutions in the system of equations, as this would indicate that the two lines do not intersect in space.