If two planes intersect, their intersection is exactly one

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if two planes intersect, their intersection is exactly one

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LectureNotes said if two planes intersect, their intersection is exactly one

Answer:
When considering the intersection of two planes in three-dimensional space, it is a fundamental concept in geometry. @username, let’s dive into this topic more comprehensively.

Detailed Explanation:

1. Understanding Planes:

  • A plane in three-dimensional space is a flat, two-dimensional surface that extends infinitely in all directions. It can be defined by a point and a normal vector, or by three non-collinear points in the space.

2. Intersection of Two Planes:

  • When two planes intersect in three-dimensional space, their intersection is always a line (except in the special case when they are parallel and do not intersect at all).

Solution By Steps:

  1. Mathematical Representation:

    • Suppose we have two planes defined by the equations:
      \mathbf{P}_1: a_1x + b_1y + c_1z = d_1
      \mathbf{P}_2: a_2x + b_2y + c_2z = d_2

  2. Intersection Condition:

    • The two planes intersect if they are not parallel. For them to interpenetrate:
      • The normal vectors of the planes (denoted as \mathbf{n}_1 = (a_1, b_1, c_1) and \mathbf{n}_2 = (a_2, b_2, c_2)) should not be multiples of each other:
        \mathbf{n}_1 \not\parallel \mathbf{n}_2
      • This implies there’s no scalar \lambda such that:
        (a_1, b_1, c_1) = \lambda (a_2, b_2, c_2)

  3. Finding the Line of Intersection:

    • The line of intersection can be found by solving the system of linear equations represented by the plane equations. The solution to these equations gives the parametric equations of the line.

      1. Step 1: Find a Direction Vector for the Line:
        • The direction vector \mathbf{d} of the line of intersection is perpendicular to both normal vectors (\mathbf{n}_1 and \mathbf{n}_2) and can be obtained by taking the cross product:
          \mathbf{d} = \mathbf{n}_1 \times \mathbf{n}_2
          • Let:
            \mathbf{d} = (a_1, b_1, c_1) \times (a_2, b_2, c_2)
            • This results in:
              \mathbf{d} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix}

  4. Step 2: Find a Point on the Line:

    • A particular solution (point of intersection, \mathbf{P}) can be found by solving the system of the plane equations for x, y, and z.

  5. Final Parametric Form of the Line:

    • Once the direction vector \mathbf{d} and a point \mathbf{P} on the line are found, the line of intersection can be expressed parametrically as:
      \mathbf{L}(t) = \mathbf{P} + t\mathbf{d}
      where t is a parameter.

Final Answer:
Therefore, if two planes intersect, their intersection is exactly one line in three-dimensional space, as evidenced by the solution steps above. This is a general geometric property of planes in three-dimensional Euclidean space.