Math Problem Statement

Find the point of intersection of the two lines L1: x = -2t, y = 1 + 2t, z = 3t and L2: x = -6 + 2s, y = 5s, z = 1 + 5s.

Solution

To find the point of intersection of the two lines L1L_1 and L2L_2, we solve the parametric equations for tt and ss such that the coordinates match:

Equations of the lines:

  1. L1:x=2t,y=1+2t,z=3tL_1: x = -2t, \, y = 1 + 2t, \, z = 3t
  2. L2:x=6+2s,y=5s,z=1+5sL_2: x = -6 + 2s, \, y = 5s, \, z = 1 + 5s

Step-by-step solution:

1. Set the xx-coordinates equal:

2t=6+2s-2t = -6 + 2s Rearrange: 2t + 2s = 6 \quad \implies \quad t + s = 3 \tag{1}

2. Set the yy-coordinates equal:

1+2t=5s1 + 2t = 5s Rearrange: 2t - 5s = -1 \tag{2}

3. Solve the system of equations (1)(1) and (2)(2):

From (1)(1): t=3st = 3 - s.

Substitute t=3st = 3 - s into (2)(2): 2(3s)5s=12(3 - s) - 5s = -1 62s5s=16 - 2s - 5s = -1 67s=1    7s=7    s=16 - 7s = -1 \quad \implies \quad 7s = 7 \quad \implies \quad s = 1

Substitute s=1s = 1 into (1)(1): t+1=3    t=2t + 1 = 3 \quad \implies \quad t = 2

4. Compute the intersection point:

Substitute t=2t = 2 into L1L_1: x=2t=2(2)=4,y=1+2t=1+2(2)=5,z=3t=3(2)=6x = -2t = -2(2) = -4, \quad y = 1 + 2t = 1 + 2(2) = 5, \quad z = 3t = 3(2) = 6

Thus, the point of intersection is: P=(4,5,6)P = (-4, 5, 6)

Would you like further clarification or additional details?


Expanding Questions:

  1. What happens if the lines do not intersect but are skew? How is this determined mathematically?
  2. How do you find the angle between two intersecting lines in three dimensions?
  3. Can parametric equations represent curves other than straight lines? Provide an example.
  4. How can this approach be extended to solve for the intersection of a line and a plane?
  5. 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