1 / 23

Mastering Vectors for Visualizing Fields & Locations in 2D & 3D

Explore vector visualization techniques, from latitude and longitude to altitude, to enhance precision in navigating space. Learn to decompose and add vectors while mastering scalar and vector products. Includes practice exercises.

wilbertr
Download Presentation

Mastering Vectors for Visualizing Fields & Locations in 2D & 3D

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. Ch.3 Vectors Visualizing Vector fields

  2. 2-D Vectors Latitude + Longitude gives location

  3. 3-D Vectors Latitude + Longitude + Altitude more precise

  4. To go from A to B Need Origin (Charlottesville), Magnitude (70.12 miles), Direction (bear SouthEast)

  5. To go from A to B 2D Map

  6. To go from A to B Decomposing a vector OR Adding several vectors

  7. Adding Vectors Lay them out head-to-tail

  8. B B Parallelogram law of Addition A + B A Head to tail: Connect first head and last tail (problem: no common reference, so need to wait!) Tail to tail: Connect diagonal of parallelogram (problem: still only do 2 at a time) Best: Decompose vectors and add components (Later)

  9. A - B Subtracting - B A

  10. A = a A |a | = 1 a = A/|A| Unit/Base Vector Vector = Direction x Magnitude (Unit Vector) (Component)

  11. Ay A = x Ax + y Ay y Ax x Unit/Base Vector y A Resolving/decomposing a vector A  (Ax, Ay) Ax = Acosq, Ay = Asinq Building/composing a vector (Ax,Ay)  A A = (Ax2 + Ay2) q = tan-1(Ay/Ax) q x

  12. B = x Bx + y By (A+B) = x(Ax+Bx) + y(Ay+By) A = x Ax + y Ay Why compose/decompose? y Ay A q y x Ax x Can treat components as scalars !! Handle all of them independently!!

  13. B q A Multiplying ? Need to take into account angle q between A and B Scalar Product A.B Vector Product A x B A.B = ABcosq

  14. Multiplying ? Need to take into account angle q between A and B Scalar Product A.B Vector Product A x B B q A A.B = ABcosq (Projection)

  15. n B q A Multiplying ? Need to take into account angle q between A and B Scalar Product A.B Vector Product A x B B q A A x B = ABsinq n (Area) Gives normal to a plane of vectors A.B = ABcosq (Projection) Gives angle between vectors

  16. n B q A Multiplying ? Need to take into account angle q between A and B Vector Product A x B A x B = ABsinq n (Area) PRACTISE THIS -- MAKE YOUR OWN MODELS !!

  17. n B q A Interchanging A and B Scalar (Dot) Product A.B Vector (Cross) Product A x B B q A A x B = - B x A A.B = B.A Orthogonal vectors have zero scalar product Parallel vectors have zero vector product

  18. = 0 = 0 = 0 = 0 = 0 = 0 = x = z = y = 1 = 1 = 1 x x x x z x z x y x y x x . x . z . z . y . y . x x x x z z z z z y y y y y x - z y x z y Coordinate Systems Helps decompose vectors and deal with scalar components x +

  19. B = x Bx + y By + z Bz A. B =Ax.Bx + Ay.By + Az.Bz A = x Ax + y Ay + z Az z y A x B = det Can see why A x B = -B x A ! x y z Ax Ay Az Bx By Bz x Using unit vectors for products

  20. Combining more vectors Scalar Triple Product A.(B x C) Vector Triple Product A x (B x C) Other combos won’t do ! A.(B.C), Ax(B.C) not defined

  21. A A C C q q B B Combining more vectors Scalar Triple Product A.(B x C) Vector Triple Product A x (B x C) Volume A.(B x C) = B.(C x A) = C.(A x B) = -A.(C x B) etc. A x (B x C) not simply related to (A x B) x C

  22. C B q A Bac-Cab Rule Vector Triple Product A x (B x C) A x (B x C) = B(A.C) – C(A.B) Bac-Cab Rule

  23. z y A. (B x C) = x Using unit vectors for products Ax Ay Az Bx By Bz Cx Cy Cz det

More Related