1 / 26

The Cross Product

The Cross Product. Third Type of Multiplying Vectors. Cross Products. 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

shona
Download Presentation

The Cross Product

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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. Using Determinants to Find Cross Products • 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

  12. Using Determinants to Find Cross Products

  13. Using Determinants to Find Cross Products

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

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

  16. Examples

  17. 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 q, where q 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. Finding a Vector Orthogonal to Two Given Vectors • 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 P1 = (0, 0, 0), • P2 = (3,-2, 1), P3 = (-1, 3, -1) and • P4 = (2, 1, 0) • Two adjacent sides of this parallelogram are u = P1P2 and v = P1P3.

  25. Finding the Area of the Parallelogram

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

More Related