1. Visualizing Vector fields
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. Parallelogram law of Addition
A + B B 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. Subtracting
A - B - B A

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

11. Unit/Base Vector y A = x Ax + y Ay Ay A q y x Ax x
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) x y q x

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

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

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

17. Interchanging A and B Scalar (Dot) Product A.B
Vector (Cross) Product A x B n A B q 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. Helps decompose vectors and deal with scalar components
Coordinate Systems
Helps decompose vectors and deal with scalar components
x . y
= 0
y . z
z . x
x . x
= 1
y . y
z . z z y
x x x
= 0
y x y
z x z
x x y
= z
y x z
= x
z x x
= y y z x - x y z x +

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

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. Combining more vectors
Scalar Triple Product
A.(B x C)
Vector Triple Product
A x (B x C) B C q A B C q A
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. Bac-Cab Rule Vector Triple Product A x (B x C)q C
Vector Triple Product
A x (B x C)
A x (B x C) = B(A.C) – C(A.B)
Bac-Cab Rule

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