1. Chapter 4 Vectors The Cardinal Directions

2. Vectors An arrow-tipped line segment used to represent different quantities. Length represents magnitude. Arrow head represents direction.

3. Vector Addition in 1 - Dimension When vectors point in the same direction we add them just as we would add any two numbers.

4. Vector Addition in 1 - Dimension When vectors point in opposite directions we subtract them just as we would with any two numbers.

6. Vector Addition in 2-Dimensions Vectors in 2-dim are added by placing the tail of one to the head of another.

7. Remember This?

8. Addition of Several Vectors The order of addition is not important. R is called the resultant.

9. Independence of Vector Quantities Perpendicular vectors can be treated independently of each other.

11. Analytical Method of Vector Addition The sum of any two vectors can be determined using trigonometry.

12. Adding Perpendicular Vectors

13. Angle θ is = a)25 deg b)14 deg c)35 deg d)45 deg

14. Angle θ is = a)25 deg b) 14 deg c) 35 deg d) 45 deg

15. Vector Components We can take two vectors and replace them with a single vector that has the same effect. This is vector addition. We can start with a single vector and think of it as a resultant of two perpendicular vectors called components. This process is called vector resolution.

16. Example

17. Example 2

18. Problem Solving Strategy In resolving vectors choose the most convenient axis according to the specifics of the problem. Choose the axis that simplifies the solution. Axis may be up-down, left-right, east-west or north-south. Be sure to specify the positive direction for each.

19. Adding Vectors at Any Angle Vector resolution is the method used. Resolve all vectors into x and y components. Add all x’s and all y’s together. Use x tot and y tot to create a right triangle. Use Pythagorean formula to calculate resultant and trig to find angle.

22. R is = ? a)15 N b)12 N c)20 N d)11N

23. R is = ? a)15 N b)12 N c)20 N d)11N

24. Θ is = ? a)53 deg b)35 deg c)25 deg d)45 deg

25. Θ is = ? a)53 deg b)35 deg c)25 deg d)45 deg

26. Applications of Vectors Vectors can be used to represent: -displacement -velocity -acceleration -force

27. Equilibrium When the net force is zero, the object is in equilibrium. When the vector sum of the forces is not zero, a force can be applied that will produce equilibrium. This force is called the equilibrant. It is equal in magnitude but opposite in direction to the resultant.

28. 3 Forces in Equilibrium: a)produce a net force. b)produce a triangle for a vector diagram. c)are called an equilibrant. d)produce an acceleration.

29. 3 Forces in Equilibrium: a)produce a net force. b)produce a triangle for a vector diagram. c)are called an equilibrant. d)produce an acceleration.

30. Gravitational Force and Inclined Planes Gravitational force always points towards center of Earth. This is weight. Choose one axis parallel to the plane and the other perpendicular to it.

31. Formulas R 2 = A 2 + B 2 – 2AB cos Θ A x = A cos Θ A Y = A sin Θ A = A x + A Y