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

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

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

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

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

Polar Coordinate System Origin and reference line are noted Point is distance r from the origin in the direction of angle , ccw from reference line Points are labeled (r,) (r,θ)

Polar to Cartesian Coordinates • Based on forming a right triangle from r and q x = r cos q y = r sin q

Cartesian to Polar Coordinates • r is the hypotenuse and q an angle q must be ccw from positive x axis for these equations to be valid

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

Example The Cartesian coordinates of a point in the xy plane are (x,y) = (-3.50, -2.50) m, as shown in the figure. Find the polar coordinates of this point. Solution:

Example The Cartesian coordinates of a point in the xy plane are (x,y) = (-3.50, -2.50) m, as shown in the figure. Find the polar coordinates of this point. Solution:

Two points in a plane have polar coordinates (2. 50 m, 30. 0°) and (3 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.

Two points in a plane have polar coordinates (2. 50 m, 30. 0°) and (3 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. (a) (b)

A skater glides along a circular path of radius 5. 00 m A skater glides along a circular path of radius 5.00 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?

A skater glides along a circular path of radius 5. 00 m A skater glides along a circular path of radius 5.00 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? (a) (b) (c)

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

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

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

A man pushing a mop across a floor causes it to undergo two displacements. The first has a magnitude of 150 cm and makes an angle of 120° with the positive x axis. The resultant displacement has a magnitude of 140 cm and is directed at an angle of 35.0° to the positive x axis. Find the magnitude and direction of the second displacement.

A man pushing a mop across a floor causes it to undergo two displacements. The first has a magnitude of 150 cm and makes an angle of 120° with the positive x axis. The resultant displacement has a magnitude of 140 cm and is directed at an angle of 35.0° to the positive x axis. Find the magnitude and direction of the second displacement. θ = 3600 – 14.70 = 345.30

As it passes over Grand Bahama Island, the eye of a hurricane is moving in a direction 60.0 north of west with a speed of 41.0 km/h. Three hours later, the course of the hurricane suddenly shifts due north, and its speed slows to 25.0 km/h. How far from Grand Bahama is the eye 4.50 h after it passes over the island?

As it passes over Grand Bahama Island, the eye of a hurricane is moving in a direction 60.0 north of west with a speed of 41.0 km/h. Three hours later, the course of the hurricane suddenly shifts due north, and its speed slows to 25.0 km/h. How far from Grand Bahama is the eye 4.50 h after it passes over the island? dtotal= Magnitude =

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.

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

Vector product = vector Rule: Angle between two vectors: Multiplying vectors: - Vector by a vector Vector product = vector (cross product) Magnitude

Direction  right hand rule Vector product Direction  right hand rule Rule: 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.

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

42: If B is added to C = 3i + 4j, the result is a vector in the positive direction of the y axis, with a magnitude equal to that of C. What is the magnitude of B?

42: If B is added to C = 3i + 4j, the result is a vector in the positive direction of the y axis, with a magnitude equal to that of C. What is the magnitude of B?

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? N D E 45º d1 d4 d3 d2

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? (a) N D E 45º d1 d4 d3 d2

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? (b) N D E 45º d1 d4 d3 (c) Return vector  negative of net displacement, D=0.57m, directed 25º South of West d2

53: Find:

53: d1perp d1 d1// θ d2

30:

30:

(b) the direction of A x B, (c) the direction of A x (B+3k)? 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

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 AxB, (c) the direction of Ax(B+3k)? ˆ y A 130º x B

39: A wheel with a radius of 45 cm rolls without sleeping along a horizontal floor. At time t1 the dot P painted on the rim of the wheel is at the point of contact between the wheel and the floor. At a later time t2, 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 displacement P during this interval?

What are (a) the magnitude and 39: A wheel with a radius of 45 cm rolls without sleeping along a horizontal floor. At time t1 the dot P painted on the rim of the wheel is at the point of contact between the wheel and the floor. At a later time t2, 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 displacement P during this interval? y d Vertical displacement: x Horizontal displacement:

Vector a has a magnitude of 5. 0 m and is directed East Vector a has a magnitude of 5.0 m and is directed East. Vector b has a magnitude of 4.0 m and is directed 35º West 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.

Vector a has a magnitude of 5. 0 m and is directed East Vector a has a magnitude of 5.0 m and is directed East. Vector b has a magnitude of 4.0 m and is directed 35º West 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. N b -a b-a a+b 125º W a E S