Chapter 3 - Vectors

1. Chapter 3 - Vectors I. Definition II. Arithmetic operations involving vectors A) Addition and subtraction - Graphical method - Analytical method  Vector components B) Multiplication

2. Definition • Vector quantity:quantity with a magnitude and a direction. It can be • represented by a vector. • Examples: displacement, velocity, acceleration. Same displacement Displacement does not describe the object’s path. Scalar quantity:quantity with magnitude, no direction. Examples: temperature, pressure

3. Vector Notation • When handwritten, use an arrow: • When printed, will be in bold print:A • When dealing with just the magnitude of a vector in print:|A| • The magnitude of the vector has physical units • The magnitude of a vector is always a positive number

4. Coordinate Systems • Used to describe the position of a point in space • Coordinate system consists of • a fixed reference point called the origin • specific axes with scales and labels • instructions on how to label a point relative to the origin and the axes

5. Cartesian Coordinate System • Also called rectangular coordinate system • x- andy- axes intersect at the origin • Points are labeled(x,y)

6. Polar Coordinate System • Origin and reference line are noted • Point is distancerfrom the origin in the direction of angle,ccwfrom reference line • Points are labeled (r,) (r,θ)

7. Polar to Cartesian Coordinates • • Based on forming a right triangle fromr andq • x = r cos q • y = r sin q

8. Cartesian to Polar Coordinates • • ris the hypotenuse andqan angle qmust beccw from positivexaxis for these equations to be valid

9. 0º<θ1<90º 90º<θ2<180º θ2 θ1 θ3 θ4 180º<θ3<270º 270º<θ4<360º Angle origin θ4=300º=-60º Review of angle reference system 90º 0º 180º Origin of angle reference system 270º

10. Example • The Cartesian coordinates of a point in thexyplane are(x,y) = (-3.50, -2.50) m, as shown in the figure. Find the polar coordinates of this point.

11. Two points in a plane have polar coordinates (2.50 m, 30.0°) and(3.80 m, 120.0°). Determine (a) the Cartesian coordinates of these points and (b) the distance between them.

12. A skater glides along a circular path of radius5.00 m. If he coasts around one half of the circle, find (a) the magnitude of the displacement vector and (b) how far the person skated. (c) What is the magnitude of the displacement if he skates all the way around the circle?

13. II. Arithmetic operations involving vectors Vector addition: - Geometrical method Rules:

14. Vector subtraction: Vector component:projection of the vector on an axis. Vector magnitude Vector direction

15. Unit vector:Vector with magnitude 1. No dimensions, no units. Vector component Vector addition: - Analytical method:adding vectors by components.

16. A man pushing a mop across a floor causes it to undergo two displacements. The first has a magnitude of150 cmand makes an angle of120°with the positive x axis. The resultant displacement has a magnitude of140 cmand is directed at an angle of35.0°to the positive x axis. Find the magnitude and direction of the second displacement.

17. As it passes over Grand Bahama Island, the eye of a hurricane is moving in a direction60.0north of west with a speed of41.0 km/h. Three hours later, the course of the hurricane suddenly shifts due north, and its speed slows to25.0 km/h. How far from Grand Bahama is the eye4.50 hafter it passes over the island?

18. Vectors & Physics: • The relationships among vectors do not depend on the location of the origin of • the coordinate system or on the orientation of the axes. • - The laws of physics are independent of the choice of coordinate system.

19. Multiplying vectors: - Vector by a scalar: - Vector by a vector: Scalar product = scalar quantity (dot product)

20. Rule: Angle between two vectors:

21. Multiplying vectors: - Vector by a vector Vector product = vector (cross product) Magnitude

22. Place a and b tail to tail without altering their orientations. • c will be along a line perpendicular to the plane that contains a and b where they meet. • 3) Sweep a into b through the small angle between them. Vector product Direction right hand rule Rule:

23. Right-handed coordinate system z k j i y x Left-handed coordinate system z k i x j y

25. Lagrange's identity The relation:

26. 42:IfBis added toC = 3i + 4j, the result is a vector in the positive direction of they axis, with a magnitude equal to that ofC. What is the magnitude ofB?

27. 50: A fire ant goes through three displacements along level ground: d1 for 0.4m SW, d2=0.5m E, d3=0.6m at 60º North of East. Let the positive x direction be East and the positive y direction be North. (a) What are the x and y components of d1, d2 and d3? (b) What are the x and the y components, the magnitude and the direction of the ant’s net displacement? (c) If the ant is to return directly to the starting point, how far and in what direction should it move? 45º d1 d2 N D E d3 d4

28. 53: Find:

29. 30:

30. 54: Vectors A and B lie in an xy plane. A has a magnitude 8.00 and angle 130º; B has components Bx= -7.72, By= -9.20. What are the angles between the negative direction of the y axis and (a) the direction of A, (b) the direction of A x B, (c) the direction of A x (B+3k)? ˆ y A 130º x B

31. 39: A wheel with a radius of45 cmrolls without sleeping along a horizontal floor. At timet1 the dotPpainted on the rim of the wheel is at the point of contact between the wheel and the floor. At a later timet2, the wheel has rolled through one-half of a revolution. What are (a) the magnitude and (b) the angle (relative to the floor) of the displacementPduring this interval?

32. Vector a has a magnitude of5.0 mand is directed East. Vectorbhas a magnitude of4.0 mand is directed35ºWest of North. What are (a) the magnitude and direction of (a+b)? (b) What are the magnitude and direction of (b-a)? (c) Draw a vector diagram for each combination.