Math Problem Statement
Solution
To find the point of intersection of the two lines and , we solve the parametric equations for and such that the coordinates match:
Equations of the lines:
Step-by-step solution:
1. Set the -coordinates equal:
Rearrange: 2t + 2s = 6 \quad \implies \quad t + s = 3 \tag{1}
2. Set the -coordinates equal:
Rearrange: 2t - 5s = -1 \tag{2}
3. Solve the system of equations and :
From : .
Substitute into :
Substitute into :
4. Compute the intersection point:
Substitute into :
Thus, the point of intersection is:
Would you like further clarification or additional details?
Expanding Questions:
- What happens if the lines do not intersect but are skew? How is this determined mathematically?
- How do you find the angle between two intersecting lines in three dimensions?
- Can parametric equations represent curves other than straight lines? Provide an example.
- How can this approach be extended to solve for the intersection of a line and a plane?
- How do you determine if two lines are parallel in three-dimensional space?
Tip:
When solving for intersections of parametric lines, always confirm the solution satisfies all equations for consistency.
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Math Problem Analysis
Mathematical Concepts
Analytic Geometry
Parametric Equations
Systems of Equations
Formulas
Equating parametric coordinates
Solving systems of linear equations
Theorems
Solution of parametric line intersection
Suitable Grade Level
Grades 10-12
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