Vector Mathematics

1. VectorMathematics Physics 1

2. Physical Quantities • A scalar quantity is expressed in terms of magnitude (amount) only. • Common examples include time, mass, volume, and temperature.

3. Physical Quantities • A vector quantity is expressed in terms of both magnitude and direction. • Common examples include velocity, weight (force), and acceleration.

4. Representing Vectors • Vector quantities can be graphically represented using arrows. • magnitude = length of the arrow • direction = arrowhead

5. Vectors • All vectors have a head and a tail.

6. Vector Addition • Vector quantities are addedgraphically by placing them head-to-tail.

7. Head-to-Tail Method • Draw the first COMPONENT vector with the proper length and orientation. • Draw the second COMPONENT vector with the proper length and orientation starting from the head of the first component vector.

8. Head-to-Tail Method • The RESULTANT (sum) vector is drawn starting at the tail of the first component vector and terminating at the head of the second component vector. • Measure the length and orientation of the resultant vector.

9. To add vectors, move tail to head and then draw resultant from original start to final point. East Resultant Resultant is (sqrt(2)) 45◦ south of East South

10. Vector addition is ‘commutative’ (can add vectors in either order) East Resultant Resultant is (sqrt(2)) 45◦ south of East South

11. Vector addition is ‘commutative’ (can add vectors in either order) East South Resultant Resultant is (sqrt(2)) 45◦ south of East Resultant South East

12. Co-linear vectors make a longer (or shorter) vector Resultant is 3 magnitude South

13. Co-linear vectors make a longer (or shorter) vector Resultant is 3 magnitude South

14. Can add multiple vectors. Just draw ‘head to tail’ for each vector Resultant is magnitude 45◦ North of East North East East North

15. Adding vectors is commutative. East North East North North Resultant is magnitude 45◦ North of East East East South

16. Equal but opposite vectors cancel each other out North West East Resultant=0 Resultant is 0. South

17. Vector Addition – same direction A + B = R A B B A R = A + B

18. Vector Addition • Example: What is the resultant vector of an object if it moved 5 m east, 5 m south, 5 m west and 5 m north?

19. Vector Addition – Opposite direction(Vector Subtraction) . A + (-B) = R A B -B A A + (-B) = R -B

20. Vectors • The sum of two or more vectors is called the resultant.

21. Practice Vector Simulator http://phet.colorado.edu/sims/vector-addition/vector-addition_en.html

22. Polar Vectors • Every vector has a magnitude and direction

23. Right Triangles SOH CAH TOA

24. Vector Resolution • Every vector quantity can be resolved into perpendicular components. • Rectilinear (component) form of vector:

25. Vector Resolution • Vector A has been resolved into two perpendicular components, Ax (horizontal component) and Ay (vertical component).

26. Vector Resolution • If these two components were added together, the resultant would be equal to vector A.

27. Vector Resolution • When resolving a vector graphically, first construct the horizontal component (Ax). Then construct the vertical component (Ay). • Using right triangle trigonometry, expressions for determining the magnitude of each component can be derived.

28. Vector Resolution • Horizontal Component (Ax)

29. Vector Resolution • Vertical Component (Ay)

30. Drawing Directions • EX: 30° S of W • Start at west axis and move south 30 ° • Degree is the angle between south and west N W E S

31. N (-,+) (+,+) E W (-,-) (+,-) S Vector Resolution • Use the sign conventions for the x-y coordinate system to determine the direction of each component.

32. Component Method • Resolve all vectors into horizontal and vertical components. • Find the sum of all horizontal components. Express as SX. • Find the sum of all vertical components. Express as SY.

33. Component Method • Construct a vector diagram using the component sums. The resultant of this sum is vector A + B. • Find the magnitude of the resultant vector A + B using the Pythagorean Theorem. • Find the direction of the resultant vector A + B using the tangent of an angle q.

34. Adding “Oblique” Vectors • Head to tail method works, but makes it very difficult to ‘understand’ the resultant vector

35. Adding “Oblique” Vectors • Break each vector into horizontal and vertical components. • Add co-linear vectors • Add resultant horizontal and vertical components

36. Adding “Oblique” Vectors • Break each vector into horizontal and vertical components.

37. Adding “Oblique” Vectors • Break each vector into horizontal and vertical components. • Add co-linear vectors

38. Adding “Oblique” Vectors • Break each vector into horizontal and vertical components. • Add co-linear vectors • Add resultant horizontal and vertical components

39. Adding “Oblique” Vectors • Break each vector into horizontal and vertical components. • Add co-linear vectors • Add resultant horizontal and vertical components

40. Using Calculator For Vectors • Can use the “Angle” button on TI-84 calculator to do vector mathematics

41. Using Calculator for Vectors