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Ch.3 Vectors

Ch.3 Vectors. Visualizing Vector fields. 2-D Vectors. Latitude + Longitude gives location. 3-D Vectors. Latitude + Longitude + Altitude more precise. To go from A to B. Need Origin (Charlottesville), Magnitude (70.12 miles), Direction (bear SouthEast). To go from A to B. 2D Map.

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Ch.3 Vectors

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  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

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