University Physics with Modern Physics Fifteenth Edition Chapter 1 Units Physical Quantities and Vectors Copyright 2020 Pearson Education Inc All Rights Reserved Vectors and Scalars A scalar quantity can be described by a single number A vector quantity has both a magnitude and a direction in space In this book a vector quantity is represented in boldface italic type with an arrow over it A is written as A or The magnitude of A A Copyright 2020 Pearson Education Inc All Rights Reserved Drawing Vectors Draw a vector as a line with an arrowhead at its tip The length of the line shows the vector s magnitude The direction of the line shows the vector s direction Copyright 2020 Pearson Education Inc All Rights Reserved Adding Two Vectors Graphically 1 of 3 Copyright 2020 Pearson Education Inc All Rights Reserved Adding Two Vectors Graphically 2 of 3 Copyright 2020 Pearson Education Inc All Rights Reserved Adding Two Vectors Graphically 3 of 3 Copyright 2020 Pearson Education Inc All Rights Reserved Adding More Than Two Vectors Graphically 1 of 3 To add several vectors use the head to tail method The vectors can be added in any order Copyright 2020 Pearson Education Inc All Rights Reserved Adding More Than Two Vectors Graphically 2 of 3 To add several vectors use the head to tail method The vectors can be added in any order Copyright 2020 Pearson Education Inc All Rights Reserved Adding More Than Two Vectors Graphically 3 of 3 To add several vectors use the head to tail method The vectors can be added in any order Copyright 2020 Pearson Education Inc All Rights Reserved Subtracting Vectors Copyright 2020 Pearson Education Inc All Rights Reserved Multiplying a Vector by a Scalar If c is a scalar the product cA has magnitude c A The figure illustrates multiplication of a vector by a a positive scalar and b a negative scalar Copyright 2020 Pearson Education Inc All Rights Reserved Addition of Two Vectors at Right Angles To add two vectors that are at right angles first add the vectors graphically Then use trigonometry to find the magnitude and direction of the sum In the figure a cross country skier ends up 2 24 km from her starting point in a direction of 63 4 degrees east of north Copyright 2020 Pearson Education Inc All Rights Reserved Components of a Vector Adding vectors graphically provides limited accuracy Vector components provide a general method for adding vectors Any vector can be represented by an x component Ax and a y component Ay Copyright 2020 Pearson Education Inc All Rights Reserved Positive and Negative Components The components of a vector may be positive or negative numbers as shown in the figures Copyright 2020 Pearson Education Inc All Rights Reserved Finding Components We can calculate the components of a vector from its magnitude and direction Video Tutor Solution Example 1 6 Copyright 2020 Pearson Education Inc All Rights Reserved Calculations Using Components We can use the components of a vector to find its magnitude and direction A y We can use the components of a A x A 2 2 and tan A y A x set of vectors to find the components of their sum R A B C x A y R y x x x Refer to Problem Solving Strategy 1 3 B C y y Copyright 2020 Pearson Education Inc All Rights Reserved Unit Vectors A unit vector has a magnitude of 1 with no units i The unit vector j points in the points in points in y direction and the x direction the z direction k Any vector can be expressed in terms of its components as A k i j A x A y A z Copyright 2020 Pearson Education Inc All Rights Reserved The Scalar Product 1 of 2 Copyright 2020 Pearson Education Inc All Rights Reserved The Scalar Product 2 of 2 The scalar product can be positive negative or zero depending on the B angle between and A Copyright 2020 Pearson Education Inc All Rights Reserved Calculating a Scalar Product Using Components In terms of components The scalar product of two vectors is the sum of the products of their respective components Copyright 2020 Pearson Education Inc All Rights Reserved Finding an Angle Using the Scalar Product Example 1 10 shows how to use components to find the angle between two vectors Copyright 2020 Pearson Education Inc All Rights Reserved The Vector Product If the vector product cross product of two vectors is C A B then The direction of the vector product can be found using the right hand rule Copyright 2020 Pearson Education Inc All Rights Reserved The Vector Product is Anticommutative A B B A Copyright 2020 Pearson Education Inc All Rights Reserved Calculating the Vector Product Use AB to find the magnitude and the right hand sin rule to find the direction Refer to Example 1 11 Video Tutor Solution Example 1 11 Copyright 2020 Pearson Education Inc All Rights Reserved