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

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

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

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

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

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

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

18. Co-linear vectors make a longer vector
East
Resultant
South

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

20. 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?

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

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

23. Practice Vector Simulator

24. Polar Vectors
Every vector has a magnitude and direction

25. Right Triangles
SOH CAH TOA

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

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

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

29. 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.

30. Vector Resolution
Horizontal Component (Ax)

31. Vector Resolution
Vertical Component (Ay)

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

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

34. 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.

35. 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.

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

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

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

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

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

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

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

43. Using Calculator for Vectors