Video 9: 3D Geometry, Angle Bisector & Line of Intersection by MIT OpenCourseWare

Summary by www.lecturesummary.com: Video 9: 3D Geometry, Angle Bisector & Line of Intersection by MIT OpenCourseWare

• • Introduction to 3D Geometry

    The video begins with an introduction covering two topics related to 3D geometry:

    • The Angle Bisector of two planes
    • The Line of Intersection of two planes (marked as more important)

    The Angle Bisector is considered an easy topic where you primarily need to memorize the formula.

    Angle Bisector of Two Planes

    The Angle Bisector is presented as a plane that divides the angle between two specified planes, P1 and P2. The acute angle and the obtuse angle between the planes are represented by two angle bisectors. To visualize planes, a notebook analogy is recommended.

    Angle Bisector Formula

    The Angle Bisector Formula is discussed, focusing on how to determine the equation for the two angle bisector planes. Finding them is very simple and involves memorizing a formula.

    Angle Bisector Formula Conditions

    Importantly, move the constant term to the left and, if required, multiply by negative to ensure D1 and D2 in the plane equations are greater than zero.

    Acute vs. Obtuse Bisector Identification

    Verify the sign of a1 a2 + b1 b2 + c1 c2.

    • The formula with the positive sign yields the obtuse angle bisector if a1 a2 + b1 b2 + c1 c2 is greater than zero.
    • The sign rule is opposite if a1 a2 + b1 b2 + c1 c2 is less than zero.

    The sign corresponding to the obtuse side is determined by the sign of a1 a2 + b1 b2 + c1 c2. The opposite sign is seen in the acute bisector.

    Angle Bisector Approach

    The main idea of this not very conceptual topic is to remember the formula and use caution when using signs and calculations if they appear in a problem. No problem-solving examples are provided for this topic.

    Line of Intersection of Two Planes

    The Line of Intersection is the very interesting part.

    Line of Intersection Concept

    The line shared by planes P1 and P2 is known as the "Line of Intersection Concept." A notebook spine is used as an analogy to help visualize this line.

    Determining the Line Equation

    You will need two things to find the equation of this line L:

    • A point on the line
    • A parallel vector

    Determining the Parallel Vector

    Both planes are parallel to the line of intersection. The B vector, which is parallel to the line, is perpendicular to both normal vectors of the planes, N1 and N2. This indicates that N1 cross N2 is the B vector. It is advised to review the definition of the cross product from the vectors chapter.

    Identifying a Point on the Line

    You must locate a point A that is shared by both planes. To accomplish this, set one coordinate to zero. Setting one of the three variables (x, y, and z) to zero reduces the system of two equations with two variables to two plane equations, which can then be solved for the point's coordinates.

    Example Problem of Line of Intersection

    The plane equations x + y + z = 1 (P1) and x - y + z = 1 (P2) are used to begin an example.

    Example Step 1

    Using the equations N1 = i + j + k and N2 = i - j + k, determine the normal vectors N1 and N2.

    Example Step 2

    Utilizing the cross product of N1 and N2, compute the B vector. The computation is displayed using the components of N1 and N2 as well as the determinant of the matrix with i, j, and k. B = 2i - 2k is the outcome. By taking the dot product, it is confirmed that this vector is perpendicular to N1 and N2.

    Step 3 Example

    Locate a point A on the line. In both plane equations, set z = 0. As a result, the system is:

    • x + y = 1
    • x - y = 1

    The results of solving these two equations are x = 1 and y = 0. Thus, (1, 0, 0) is point A.

    Line Equation Example

    Using point A and parallel vector B, write the line's equation:

    • Vector form: r = (i + 0j + 0k) + lambda (2i + 0j - 2k), which simplifies to r = i + lambda (2i - 2k).
    • Coordinate form: (x - 1) / 2 = (y - 0) / 0 = (z - 0) / -2. y = 0 is implied by the term (y - 0) / 0.

    Summary of the Line of Intersection

    Remember that:

    • A point can be located by setting one coordinate to zero.
    • The line of intersection is common to both planes.
    • Its parallel vector is N1 cross N2.

    Conclusion

    The angle bisector and line of intersection of two planes were discussed in the video. Lines, planes, points, etc., will all be mixed in the upcoming video.