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Cross Product Calculus: Operations, Properties, and Applications

Explore scalar and vector operations, including scalar multiplication, vector addition, subtraction, dot and cross products. Learn determinant definitions, properties, and examples. Discover applications like finding area and volume using cross products and triple scalar products. Understand coplanarity between vectors and points.

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Cross Product Calculus: Operations, Properties, and Applications

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  1. Section 11.4The Cross Product Calculus III September 22, 2009 Berkley High School

  2. Operations on Vectors(So Far) • Scalar Multiplication • Vector Addition • Vector Subtraction • Dot Product

  3. New Operation: Cross Product

  4. Definition of Determinant

  5. Definition of Determinant

  6. Definition of Cross Product

  7. Example

  8. Example

  9. Properties: Direction & Magnitude

  10. Properties: Direction & Magnitude

  11. Properties: Area of Parallelogram

  12. Properties

  13. Assignment • Section 11.4, 1-35, odd x25 • But wait, there’s more…

  14. Volume of Parallelepiped

  15. Volume of Parallelepiped

  16. Volume of Parallelepiped

  17. Volume of Parallelepiped

  18. Volume of Parallelepiped

  19. Volume of Parallelepiped

  20. Volume of Parallelepiped Triple Scalar Product

  21. Example

  22. Example

  23. Are two vectors coplanar? • Any two non-zero vectors define a plane. • Are three vectors coplanar? • The vectors are coplanar iff the parallelepiped formed by the three vectors has Volume=0

  24. Example

  25. Are three points coplanar? • Any three points define a plane. • Are four points coplanar? • From the four points, three vectors can be formed with a common tail. If the vectors are coplanar, then the points are coplanar.

  26. Assignment • 11.4, 41-47 odd

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