Chapter 3 Vectors and Coordinate Systems Our universe

Chapter 3. Vectors and Coordinate Systems Our universe has three dimensions, so some quantities also need a direction for a full description. For example, wind has both a speed and a direction; hence the motion of the wind is described by a vector. Chapter Goal: To learn how vectors are represented and used. Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Student Learning Objectives – Ch. 3 • To understand the basic properties of vectors. • To add and subtract vectors both graphically and using components. • To be able to decompose a vector into its components and to reassemble vector components into a magnitude and a direction. • To recognize and use the basic unit vectors. • To work with tilted coordinate systems. Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Graphical Vector Addition Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Tip to Tail Method Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Parallelogram Method Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Vector Addition Problem • Which figure shows A 1 + A 2 + A 3? Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Which figure shows Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley. ?

Multiplication by a scalar Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Vector Subtraction Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Vector Subtraction • Which figure shows 2 A – B? Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Which figure shows 2 − Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley. ?

Components of vectors Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Magnitude of A: A = (Ax 2 + Ay 2)1/2 Direction of A: θ = tan-1 (Ay/Ax) Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

What are the x- and y-components Cx and Cy of vector ? A. B. C. D. E. Cx = 1 cm, Cy = – 1 cm Cx = – 3 cm, Cy = 1 cm Cx = – 2 cm, Cy = 1 cm Cx = – 4 cm, Cy = 2 cm Cx = – 3 cm, Cy = – 1 cm Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

What are the x- and y-components Cx and Cy of vector ? A. B. C. D. E. Cx = 1 cm, Cy = – 1 cm Cx = – 3 cm, Cy = 1 cm Cx = – 2 cm, Cy = 1 cm Cx = – 4 cm, Cy = 2 cm Cx = – 3 cm, Cy = – 1 cm Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Workbook problems 12, 13, 15, 16, 18 Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Workbook problems 12, 13, 15, 16, 18 answers Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Workbook exercises 25 -29 Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Workbook exercises 25 -29 - answers Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Tilted axes • Often is it convenient to tilt the coordinate axes (to represent an object on an incline for example). • The axes stay perpendicular to each other. • The unit vectors corespond to axes, not to “horizontal and vertical” so they are also tilted. Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Tilted axes • Cx = C cos θ • Cy = C sin θ • Note that θ is defined relative to the tilted x-axis and not to “horizontal” Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

EXAMPLE 3. 7 Finding the force perpendicular to a surface Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

EXAMPLE 3. 7 Finding the force perpendicular to a surface Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

EXAMPLE 3. 7 Finding the force perpendicular to a surface Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Workbook problems 26, 27, 28, 30, 31 Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Chapter 3. Summary Slides Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Important Concepts Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Important Concepts Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Using Vectors Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Using Vectors Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Using Vectors Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Using Vectors Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Chapter 3. Clicker Questions Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Which figure shows Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley. ?

Which figure shows 2 − Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley. ?

Angle φ that specifies the direction of is given by A. tan– 1(Cy /Cx) B. tan– 1(Cx /|Cy|) C. tan– 1(Cy /|Cx|) D. tan– 1(Cx /Cy) E. tan– 1(|Cx |/|Cy|) Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.

Angle φ that specifies the direction of is given by A. tan– 1(Cy /Cx) B. tan– 1(Cx /|Cy|) C. tan– 1(Cy /|Cx|) D. tan– 1(Cx /Cy) E. tan– 1(|Cx |/|Cy|) Copyright © 2008 Pearson Education, Inc. , publishing as Pearson Addison-Wesley.