1. The Cross Product Third Type of Multiplying Vectors

2. Cross Products

3. Determinants It is much easier to do this using determinants because we do not have to memorize a formula. Determinants were used last year when doing matrices Remember that you multiply each number across and subtract their products

4. Finding Cross Products Using Equation

5. Evaluating a Determinant

6. Evaluating Determinants

7. Using Determinants to Find Cross Products This concept can help us find cross products. Ignore the numbers included in the column under the vector that will be inserted when setting up the determinant.

8. Using Determinants to Find Cross Products Find v x w given v = i + j w = 2i + j + k

9. Using Determinants to Find Cross Products

10. If v = 2i + 3j + 5k and w = i + 2j + 3k, find (a) v x w (b) w x v (c) v x v

11. Using Determinants to Find Cross Products

14. Algebraic Properties of the Cross Product If u, v, and w are vectors in space and if  is a scalar, then u x u = 0 u x v = -(v x u)  (u x v) = (  u) x v = u x (  v) u x (v + w) = (u x v) + (u x w)

15. Examples Given u = 2i – 3j + k v = -3i + 3j + 2k w = i + j + 3k Find (a) (3u) x v (b) v. (u x w)

16. Examples

18. Geometric Properties of the Cross Product Let u and v be vectors in space u x v is orthogonal to both u and v. ||u x v|| = ||u|| ||v|| sin  where  is the angle between u and v. ||u x v|| is the area of the parallelogram having u ≠ 0 and v ≠ 0 as adjacent sides

19. Geometric Properties of the Cross Product u x v = 0 if and only if u and v are parallel.

20. Finding a Vector Orthogonal to Two Given Vectors Find a vector that is orthogonal to u = 2i – 3j + k and v = i + j + 3k According to the preceding slide, u x v is orthogonal to both u and v. So to find the vector just do u x v

21. Finding a Vector Orthogonal to Two Given Vectors

22. To check to see if the answer is correct, do a dot product with one of the given vectors. Remember, if the dot product = 0 the vectors are orthogonal

23. Finding a Vector Orthogonal to Two Given Vectors

24. Finding the Area of a Parallelogram Find the area of the parallelogram whose vertices are P 1 = (0, 0, 0), P 2 = (3,-2, 1), P 3 = (-1, 3, -1) and P 4 = (2, 1, 0) Two adjacent sides of this parallelogram are u = P 1 P 2 and v = P 1 P 3.

25. Finding the Area of the Parallelogram

26. Your Turn Try to do page 653 problems 1 – 47 odd.