1. Vectors Part Trois

2. Vector Geometry OQ is known as the resultant of a and b
Consider this parallelogram Q O P R a b
Opposite sides are Parallel
OQ is known as the resultant of a and b

3. Resultant of Two Vectors
Is the same, no matter which route is followed
Use this to find vectors in geometrical figures

4. . Example = a + ½b S is the Midpoint of PQ. Work out the vector Q S P

5. . Alternatively = b + a - ½b = ½b + a = a + ½b
S is the Midpoint of PQ.
Work out the vector Q O P R a b . S
= b
+ a
- ½b
= ½b + a
= a + ½b

6. Example AC= p, AB = q M is the Midpoint of BC Find BC BC BA AC = +
= -q + p
= p - q

7. Example AC= p, AB = q M is the Midpoint of BC Find BM BM ½BC =
= ½(p – q)

8. Example AC= p, AB = q M is the Midpoint of BC Find AM AM + ½BC = AB
= q + ½(p – q)
= q +½p - ½q
= ½q +½p
= ½(q + p)
= ½(p + q)

9. Alternatively AC= p, AB = q M is the Midpoint of BC Find AM AM + ½CB =
= p + ½(q – p)
= p +½q - ½p
= ½p +½q
= ½(p + q)

10. Why?? We represent objects using mainly linear primitives:
points
lines, segments
planes, polygons
Need to know how to compute distances, transformations…

11. Basic definitions Points specify location in space (or in the plane).
Vectors have magnitude and direction (like velocity).
Points  Vectors

12. Point + vector = point

13. vector + vector = vector
Parallelogram rule

14. point - point = vector
Final - Initial
B – A B A
A – B B A

15. point + point: not defined!!
Makes no sense

16. Map points to vectors If we have a coordinate system with
origin at point O
We can define correspondence between points and vectors:

17. Position Vectors
A position vector for a point X (with respect to the origin 0) is the fixed vector X O

18. Components of a Vector Between Two Points
Given and , then
Given and , then
Proof:

19. Understanding

20. Understanding

21. Understanding

22. Understanding
Determine the coordinates of the point that divides AB in the ratio 5:3 where A is (2, -1, 4) and B is (3, 1, 7)
If AP:PB = m:n then

23. Understanding
Consider collinear points P, Q, and R. consider also a reference point O. Write OQ as a linear combination of OP and OR if:
Q divides PR (internally) in the ratio 3:5
Q divides PR (externally) in the ratio -2:7
Since:
Then: P Q R 3 5

24. Understanding
Consider collinear points P, Q, and R. consider also a reference point O. Write OQ as a linear combination of OP and OR if:
Q divides PR (internally) in the ratio 3:5
Q divides PR (externally) in the ratio -2:7
Since:
Then: 7 Q P R -2 5

25. Understanding
Determine the point Q that divides line segment PR in the ratio 5:8 if P is (1,5,2) and R is (-4,1,2).

26. Inner (dot) product Defined for vectors: w  v L
w·cos(θ) is the scalar projection of w on vector v, which we will call L w
the dot product can be understood geometrically as the product of the length of this projection and the length of v.  v L
The dot product or scalar product allows us to determine a product the magnitude of one vector by the magnitude of the component of the other vector that’s parallel to the first. (The dot product was invented specifically for this purpose.)

27. Dot product in coordinatesy yw
An interesting proof w yv v xw xv O x

28. Proof of Geometric and Coordinate Definitions Equality
From the Cosine Law
therefore

29. Vector Properties

30. Understanding

31. Understanding
Prove that for any vector ,
Proof:

32. Understanding
Cosine has the range of -1 to 1

33. Understanding
Given that work is defined as (the dot product of force, f, and distance travelled, s. A crate on a ramp is hauled 8 m up the ramp under a constant force of 20 N applied at an angle of 300 to the ramp. Determine the work done.

34. Understanding
The angle between two vectors u and v is 1100, if the magnitude of vector u is 12, then determine:
The projection of vector u onto vector v

35. The Cross Product
Some physical concepts (torque, angular momentum, magnetic force) require that we multiply the magnitude of one vector by the magnitude of the component of the other vector that is perpendicular to the first (recall in the dot product we used the parallel component).
The cross product was invented for this specific purpose
Bsin(θ)
The magnitude (length) of the cross product A B θ

36. Unlike the dot product, the cross product is a vector quantity.
Where θ is the angle between and , and is a unit vector perpendicular to both and such that and form a right handed triangle.
Take your right hand, and point all 4 fingers (not the thumb) in the direction of the first vector x. Next, rotate your arm or wrist so that you can curl your fingers in the direction of y. Then extend your thumb, which is perpendicular to both x and y. The thumb’s direction is the direction the cross product will point.

37. Understanding
Out off board
Into board

38. Understanding
First suppose u and v are collinear, then the angle, θ, between u and v is 00. Since sin(00)=0, then:
Conversely suppose that then Since u and v are non-zero vectors, then sin(θ)=0, and hence θ=0 or θ=π.
Therefore u and v are collinear.

39. Properties

40. Understanding
Determine the area of the parallelogram below, using the cross product. 8
300 13

41. Understanding
State whether each expression has meaning. If not, explain why, if so, state whether it is a vector or a scalar quantity.
scalar
Non sense
vector
Non sense
vector
Non sense
scalar

42. Understanding
A right-threaded bolt is tightened by applying a 50 N force to a 0.20 m wrench as shown in the diagram. Find the moment of the force about the centre of the bolt.
The moment of the bolt is 9.4 Nm and is directed down

43. Vector Product in Component Form
Recall:
But how do we obtain the answer in component form?
The easiest technique is called the Sarrus’ Scheme

44. Understanding

45. Understanding A D B C a.

46. Understanding
Area of triangle is one-half the area of the parallelogram b.

47. Understanding

48. Understanding Find a unit vector that is perpendicular to both:
We need to apply the cross product to determine a vector perpendicular to both u and v. Then we divide by the length of this vector to make it a unit vector

49. Useful Properties