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

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

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

Coordinate Systems Used to describe the position of a point in spaceUsed to describe the position of a point in space Coordinate system consists ofCoordinate 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 systemAlso called rectangular coordinate system x- and y- axes intersect at the originx- and y- axes intersect at the origin Points are labeled (x,y)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 Based on forming a right triangle from r and  x = r cos x = r cos  y = r sin y = r sin 

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

Review of angle reference system Origin of angle reference system θ1θ1 0º<θ 1 <90º 90º<θ 2 <180º θ2θ2θ2θ2 180º<θ 3 <270º θ3θ3θ3θ3 θ4θ4θ4θ4 270º<θ 4 <360º 90º 180º 270º 0º0º0º0º θ 4 =300º=-60º Angle origin

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.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.

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 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?

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

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

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

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.

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?

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 coordinate system or on the orientation of the axes. - The laws of physics are independent of the choice of coordinate system - The laws of physics are independent of the choice of coordinate system.

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

Rule: Angle between two vectors:

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

Rule: Direction  right hand rule 1)Place a and b tail to tail without altering their orientations. 2)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

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

Lagrange's identity The relation:

: 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?

A fire ant goes through three displacements along level ground: d 1 for 0.4m SW, d 2 =0.5m E, d 3 =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 d 1, d 2 and d 3 ? (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? 50: A fire ant goes through three displacements along level ground: d 1 for 0.4m SW, d 2 =0.5m E, d 3 =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 d 1, d 2 and d 3 ? (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 E d3d3 d2d2 45º d1d1 D d4d4

53: Find:

30 :

y x A B 130º 54 : Vectors A and B lie in an xy plane. A has a magnitude 8.00 and angle 130º; B has components B x = -7.72, B y = 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

A wheel with a radius ofrolls without sleeping along a horizontal floor. At timethe dotpainted on the rim of the wheel is at the point of contact between the wheel and the floor. At a later time, 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 displacementduring this interval? 39 : A wheel with a radius of 45 cm rolls without sleeping along a horizontal floor. At time t 1 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 t 2, 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?

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.