Emath lecture recording on 26 February 2025 at 08.52 AM

Volume of a Parallelepiped

• To find the volume of a parallelepiped, the formula is:

  • Volume = Area of Base × Perpendicular Height

• Base Shape: The base is a parallelogram defined by two vectors, y and z.

• Area of the Base:

  • Area can be computed using vector methods as:

    • Area = ||y × z|| (the length of the cross product of vectors y and z).

• Perpendicular Height:

  • To find the height, we need a vector orthogonal to the base,

    • This vector can be obtained by using the cross product of y and z:

      • Normal Vector = y × z.

  • The height can be calculated by projecting another vector x onto the orthogonal vector (the vector obtained from y × z).

Projection Formula and Volume Calculation

• Projection of Vector: To find the height, we project vector x onto the vector (y × z).

• The projection formula is given by:

  • Projection = (x · (y × z)) / ||y × z||² (y × z)

• By utilizing this projection:

  • We can obtain the required height value that allows us to compute volume.

• The volume can thus be simplified to:

  • Volume = |x · (y × z)| (absolute value of the scalar triple product of vectors x, y, and z).

Scalar Triple Product

• Definition: The scalar triple product is defined as:

  • Scalar Triple Product = x · (y × z).

• Why ‘Scalar’ and ‘Triple’:

  • It’s termed scalar because the outcome is scalar (a single number).

  • It's triple due to involving three vectors (x, y, z).

• Geometric Interpretation: The absolute value of the scalar triple product gives the volume of the parallelepiped formed by the three vectors.

Applications: Line and Plane Intersections

• Determining Intersection:

  • Given a line defined by a vector equation and a plane defined by a scalar equation, ascertain if the line intersects the plane.

• Procedure:

  1. Convert the vector equation of the line into scalar parametric equations.

  2. Substitute these into the plane equation, creating simultaneous equations.

  3. Solve for the parameter (usually denoted as s) to find intersection point coordinates.

Cases of Line-Plane Interaction

1. Intersection: Leads to one solution, giving coordinates of intersection point.

2. Parallel: No intersection, yielding unsolvable equations (e.g., a statement like 0s = nonzero).

3. Embedded: The line lies within the plane, yielding infinitely many solutions (every point on the line is a solution).

Intersections of Two Lines

• Check for Intersections:

  • In 3-dimensional space, two lines might not always intersect.

• Skew Lines: Two non-parallel lines that do not intersect are called skew lines.

• Equating Lines: Set the vector parameter equations equal, leading to simultaneous equations.

• Geometric Outcomes:

  • If the solution to simultaneous equations yields different parameters for intersection, the lines do not intersect.

  • If the lines share the same parameter across the equations, they intersect at a single point.

Summary of Solutions for Line and Plane Intersections

• Three outcomes are possible when analyzing the intersection of a line with a plane or between two lines in space:

  • 1. One intersection point.

  • 2. Infinite intersection points (entire line on the plane).

  • 3. No intersection (parallel lines).

• Visual aids and clear sketches help in understanding geometrical relationships in space.