Vectors. both a magnitude and a direction. Slide Pearson Education, Inc.

Transcription

1 Vectors A quantity that is fully described The velocity vector has both a magnitude and a direction. by a single number is called a scalar quantity (i.e., mass, temperature, volume). A quantity having both a magnitude and a direction is called a vector quantity. The geometric representation of a vector is an arrow with the tail of the arrow placed at the point where the measurement is made. We label vectors by drawing a small arrow over the letter that represents the vector, i.e.,: for position, for velocity, for acceleration. Slide 3-16

2 Properties of Vectors Suppose Sam starts from his front door, takes a walk, and ends up 200 ft to the northeast of where he started. We can write Sam s displacement as: The magnitude of Sam s displacement is S S 200 ft, the distance between his initial and final points. Slide 3-17

3 Properties of Vectors Sam and Bill are neighbors. They both walk 200 ft to the northeast of their own front doors. Bill s displacement B (200 ft, northeast) has the same magnitude and direction as Sam s displacement S. Two vectors are equal if they have the same magnitude and direction. This is true regardless of the starting points of the vectors. B S. Slide 3-18

4 Terminology for comparing vectors

5 Scalar Multiplication In general, when a vector is multiplied by a positive scalar, the result is a new vector that is parallel to the original. In general, when a vector is multiplied by a negative scalar, the result is a new vector that is antiparallel to the original.

6 1-D Vector Addition and Subtraction 1-D Vector addition has the effect of increasing a vector s length 1-D Vector subtraction has the effect of decreasing a vector s length and in some cases reversing the direction.

7 2-D Vector addition Slide 1-33

8 QuickCheck 1.3 Given vectors and, what is? Slide 1-38

9 2-D Vector subtraction Slide 1-36

10 QuickCheck 1.4 Given vectors and, what is? Slide 1-40

11 Algebraic Properties Commutative Property We can add vectors in any order. Associative Property We can group vectors in any order.

12 Coordinate Systems and Vector Components A coordinate system is an artificially imposed grid that you place on a problem. You are free to choose: Where to place the origin, and How to orient the axes. Below is a conventional xy-coordinate system and the four quadrants I through IV. The navigator had better know which way to go, and how far, if she and the crew are to make landfall at the expected location. Slide 3-29

13 Component Vectors The figure shows a vector A and a standard xy-coordinate system. We can define two new vectors that we call the component vectors of A, such that: The component vectors of A are perpendicular to each other and parallel to their respective coordinate axes. This is called the decomposition of A into its component vectors. Slide 3-30

14 Components Suppose a vector A has been decomposed into component vectors A x and A y parallel to the coordinate axes. We can describe each component vector with a single number called the component. The component tells us how big the component vector is, and, with its sign, which ends of the axis the component vector points toward. Shown to the right are two examples of determining the components of a vector. Slide 3-31

15 QuickCheck 3.3 What are the x- and y-components of this vector? A. 3, 2 B. 2, 3 C. 3, 2 D. 2, 3 Slide 3-34

16 QuickCheck 3.4 What are the x- and y-components of this vector? A. 3, 4 B. 3, 4 C. 4, 3 D. 3, 4 Slide 3-36

17 QuickCheck 3.5 What are the x- and y-components of vector C? A. 1, 3 B. 3, 1 C. 1, 1 D. 4, 2 Slide 3-38

18 Moving Between the Geometric Representation and the Component Representation We will frequently need to decompose a vector into its components. We will also need to reassemble a vector from its components. The figure to the right shows how to move back and forth between the geometric and component representations of a vector. Slide 3-39

19 Moving Between the Geometric Representation and the Component Representation If a component vector points left (or down), you must manually insert a minus sign in front of the component, as done for B y in the figure to the right. The role of sines and cosines can be reversed, depending upon which angle is used to define the direction. The angle used to define the direction is almost always between 0 and 90. Slide 3-40

20 QuickCheck 3.7 The angle Φ that specifies the direction of vector is A. tan 1 (C x /C y ). B. tan 1 (C y /C x ). C. tan 1 ( C x / C y ). D. tan 1 ( C y / C x ). Slide 3-59

21 Unit Vectors Each vector in the figure to the right has a magnitude of 1, no units, and is parallel to a coordinate axis. A vector with these properties is called a unit vector. These unit vectors have the special symbols: Unit vectors establish the directions of the positive axes of the coordinate system. Slide 3-45

22 Vector Algebra When decomposing a vector, unit vectors provide a useful way to write component vectors: The full decomposition of the vector A can then be written: Slide 3-46

23 QuickCheck 3.6 Vector C can be written A. 3î + ĵ. B. 4î + 2ĵ. C. î 3ĵ. D. 2î 4ĵ. Slide 3-47

24 Example 3.4 Finding the Direction of Motion Slide 3-43

25 Working With Vectors We can perform vector addition by adding the x- and y- components separately. This method is called algebraic addition. For example, if D A B C, then: Similarly, to find R P Q we would compute: To find T cs, where c is a scalar, we would compute: Slide 3-51

26 Example 1 Vectors with Components Let A = 4î 2ĵ, B = -3î + 5ĵ, and C = A + B. a) Write C in component form b) Draw a coordinate system and on it show vectors A, B, and C. c) What is the magnitude and direction of C?

27 Example 2 Vector Algebra Find vector B if A + B + C = ĵ Find both magnitude and direction, and component form of B.