MAT 1236 Calculus III

1. MAT 1236Calculus III Section 12.4 The Cross Product http://myhome.spu.edu/lauw

2. HW… • WebAssign 12.4 • Read 12.5 (Seriously!): The first not too easy section in Calculus

3. Preview • Define a new operation on vectors: The Cross Product • Unlike the dot product, the cross product of two vectors is a vector. • Properties of the cross product.

4. The Right Hand Rule • FBI

5. We are Interested in … • Given 2 vectors, they “span” a plane • Find a vector perpendicular to this plane

6. The Cross Product If and , the cross product of a and b is the vector

7. The Cross Product The formula is traditionally memorized by using (formal) determinant expansions

8. 2x2 Determinant Expansions

9. 3x3 Determinant Expansions

10. 3x3 Determinant Expansions

11. 3x3 Determinant Expansions

12. 3x3 Determinant Expansions

13. The Cross Product The formula is traditionally memorized by using (formal) determinant expansions

14. Example 1

15. Expectations • You are expected to use the above standard procedure to find the cross product. • You are expected to show all the steps. Keep in mind, good practices are key to minimize the chance of making mistakes.

16. Property A

17. Property B

18. Property B In addition, the cross product obeys the Right Hand Rule.

19. Property B (Why?)

20. Example 1 (Verify Property B)

21. Property C

22. Property C (Why?)

23. In Particular

24. In Particular is in the same direction of k and

25. Property D Two nonzero vectors and are parallel if and only if

26. Property D (Why?) Two nonzero vectors and are parallel if and only if

27. Property E The length of the cross product axb is equal to the area of the parallelogram determined by a and b.

28. Example 2 Find a vector perpendicular to the plane that passes through the points P(6,0,0) , Q(1,1,1), R(0,0,2)

29. Example 3 Find the area of the triangle with vertices P(6,0,0) , Q(1,1,1), R(0,0,2)

30. Other Properties Right Hand Rule Default Reference only