Answer
See the explanation
Work Step by Step
To find the intersection of two lines in space, you need to determine if they intersect at a single point, are parallel and do not intersect, or are coincident (meaning they lie on top of each other).
1. Intersection of Two Lines in Space:
- If the lines intersect at a single point, you can find the coordinates of the intersection point by solving the system of equations formed by the parametric equations of the two lines.
- For example, consider the following lines:
Line 1: $x = 2 + t$, $y = 3 - t$ ,$ z = 1 + 2t$
Line 2: $x = 4 - 2s$ , $y = 1 + 3s$ , $z = 5 - s$
By equating the corresponding components of the two lines, you can solve for the values of t and s, which will give you the coordinates of the intersection point.
2. Intersection of a Line and a Plane:
- To find the intersection of a line and a plane, you need to determine if the line lies on the plane, is parallel to the plane and does not intersect, or intersects the plane at a single point.
- For example, consider the following line and plane:
Line: $x = 2 + t$, $y = 3 - t$ , $ z = 1 + 2t$
Plane: $2x + 3y - z = 7$
By substituting the parametric equations of the line into the equation of the plane, you can solve for the value of t. If a valid value of t is obtained, it means the line intersects the plane at a single point.
3. Intersection of Two Planes:
- To find the intersection of two planes, you need to determine if they intersect along a line, are parallel and do not intersect, or coincide (meaning they are the same plane).
- For example, consider the following planes:
Plane 1: $2x + 3y - z = 7$
Plane 2: $x - 2y + 4z = 5$
By setting the equations of the two planes equal to each other, you can solve for the values of $x$, $y$, and $z$. If a valid solution is obtained, it means the planes intersect along a line.
It's important to note that in some cases, the lines or planes may not intersect or coincide, resulting in no solution.
