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.

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.

Graphical Vector Addition

Tip to Tail Method

Parallelogram Method

Vector Addition Problem Which figure shows A1 + A2 + A3?

Which figure shows ? STT3.1

Multiplication by a scalar

Vector Subtraction

Vector Subtraction Which figure shows 2A – B?

Which figure shows 2 − ? STT3.2

Components of vectors

Magnitude of A: A = (Ax2 + Ay2)1/2 Direction of A: θ = tan-1 (Ay/Ax)

What are the x- and y-components Cx and Cy of vector ? 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 Answer D

What are the x- and y-components Cx and Cy of vector ? 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 STT3.3

Workbook problems 12, 13, 15, 16, 18

Workbook problems 12, 13, 15, 16, 18 - answers

Workbook exercises 25-29

Workbook exercises 25-29 - answers

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.

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”

EXAMPLE 3.7 Finding the force perpendicular to a surface

EXAMPLE 3.7 Finding the force perpendicular to a surface

EXAMPLE 3.7 Finding the force perpendicular to a surface

Workbook problems 26, 27,28,30, 31

Chapter 3. Summary Slides

Important Concepts

Important Concepts

Using Vectors

Using Vectors

Using Vectors

Using Vectors

Chapter 3. Clicker Questions

Which figure shows ? Answer C

Which figure shows 2 − ? Answer A

Angle φ that specifies the direction of is given by tan–1(Cy /Cx) tan–1(Cx /|Cy|) tan–1(Cy /|Cx|) tan–1(Cx /Cy) tan–1(|Cx |/|Cy|) Answer D

Angle φ that specifies the direction of is given by tan–1(Cy /Cx) tan–1(Cx /|Cy|) tan–1(Cy /|Cx|) tan–1(Cx /Cy) tan–1(|Cx |/|Cy|) STT3.4