1. Slide 7.3 - 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Vectors Learn to represent vectors geometrically. Learn to represent vectors algebraically. Learn the definition of a unit vector. Learn to write a vector in terms of its magnitude and direction. Learn the definition of the dot product. Learn to find the angle between two vectors. Learn the definition of work. SECTION 7.3 1 2 3 4 5 6 7

3. Slide 7.3 - 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley VECTORS Many physical quantities, such as length, area, volume, mass, and temperature, are completely described by their magnitudes in appropriate units. Such quantities are called scalar quantities. Other physical quantities, such as velocity, acceleration, and force, are not completely determined until both a magnitude (size) and a direction are specified. For example, the movement of wind is usually described by its speed (magnitude) and the direction. The wind speed and wind direction together form a vector quantity called the wind velocity.

4. Slide 7.3 - 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley GEOMETRIC VECTORS A vector can be represented geometrically by a line segment with an arrowhead. The direction of the arrow specifies the direction of the vector. The length of the arrow describes its magnitude. The tail of the arrow is called the initial point of the vector, and the tip of the arrow the terminal point. We shall denote vectors by lowercase boldface type, such as a, b, i, j, u, v, and w. When discussing vectors, we refer to real numbers as scalars. Scalars will be denoted by lower case italic type, such as a, b, x, y, and z.

5. Slide 7.3 - 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley GEOMETRIC VECTORS If the initial point of a vector v is P and the terminal point is Q, we write The magnitude (or norm) of a vector denoted by is the length of the vector v and is a scalar quantity.

6. Slide 7.3 - 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EQUIVALENT VECTORS Two vectors having the same length and same direction are called equivalent vectors.

7. Slide 7.3 - 7 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EQUIVALENT VECTORS Equivalent vectors are regarded as equal even though they may be located in different positions.

8. Slide 7.3 - 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley ZERO VECTOR The vector of length zero is called the zero vector and is denoted by 0. The zero vector has zero magnitude and arbitrary direction. If vectors v and a, have the same length and opposite direction, then a is the opposite vector of v and we write a = –v.

9. Slide 7.3 - 9 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley VECTOR ADDITION (TRIANGLE RULE) Let v and w be any two vectors. Place the vector w so that its initial point coincides with the terminal point of v. The vector v + w, called the resultant vector, is represented by the arrow from the initial point of v to the terminal point of w. v wv + w

10. Slide 7.3 - 10 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley VECTOR ADDITION (PARALLELOGRAM RULE) Place vectors v and w so that their initial points coincide. Form a parallelogram with v and w as the adjacent sides. The vector with the same initial point as the initial points of v and w that coincides with the diagonal of the parallelogram represents the resultant vector, v + w. Note that: v + w = w + v

11. Slide 7.3 - 11 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley VECTOR SUBTRACTION For any two vectors v and w, v – w = v + (–w), where –w is the opposite of w.

12. Slide 7.3 - 12 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley SCALAR MULTIPLES OF VECTORS Let v be a vector and c a scalar (a real number). The vector cv is called the scalar multiple of v. If c > 0, cv has the same direction as v and magnitude c||v||. If c < 0, cv has the opposite direction as v and magnitude |c| ||v||. If c = 0, cv = 0v = 0.

13. Slide 7.3 - 13 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Geometric Vectors Use the vectors u, v, and w to graph each vector. a. u – 2w b. 2v – u + w Solution a. u – 2w

14. Slide 7.3 - 14 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Geometric Vectors Solution continued b. 2v – u + w

15. Slide 7.3 - 15 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley ALGEBRAIC VECTORS Specifying the terminal point of the vector will completely determine the vector. For the position vector v with initial point at the origin O and terminal point at P(v 1, v 2 ), we denote the vector by A vector drawn with its initial point at the origin is called a position vector.

16. Slide 7.3 - 16 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley COMPONENTS OF VECTORS A point is denoted (v 1, v 2 ), We call v 1 and v 2 the components of the vector v; v 1 is the first component, and v 2 is the second component. we denote a vector The magnitude of follows directly from the Pythagorean Theorem,

17. Slide 7.3 - 17 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EQUIVALENT VECTORS points must coincide. If equivalent vectors, v and w, are located so that their initial are at the origin, then their terminal

18. Slide 7.3 - 18 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley REPRESENTING A VECTOR AS A POSITION VECTOR The vector with initial point P(x 1, y 1 ) and terminal point Q(x 2, y 2 ) is equal to the position vector

19. Slide 7.3 - 19 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Representing a Vector in the Cartesian Plane Let v be the vector with initial point P(4, –2) and terminal point Q(–1, 3). Write v as a position vector. Solution v has: initial point P(4, –2), so x 1 = 4 and y 1 = –2 terminal point Q(–1, 3), so x 2 = –1 and y 2 = 3 Thus,

20. Slide 7.3 - 20 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley ARITHMETIC OPERATIONS ON VECTORS are vectors and c is any scalar, then

21. Slide 7.3 - 21 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Operations on Vectors Find each expression. a. v + w b. –2v c. 2v – w d. ||2v – w|| Solution

22. Slide 7.3 - 22 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Operations on Vectors Solution continued

23. Slide 7.3 - 23 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley UNIT VECTORS A unit vector has length 1. The unit vector in the same direction as v is given by

24. Slide 7.3 - 24 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Finding a Unit Vector Find a unit vector u in the direction of Solution First, find the magnitude of Now, Check:

25. Slide 7.3 - 25 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley UNIT VECTORS i AND j In a Cartesian coordinate plane, two unit vectors that lie along the coordinate axes are particularly important. These are the vectors A vector v from (0, 0) to (v 1, v 2 ) can be represented in the form with

26. Slide 7.3 - 26 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Vectors Involving i and j Find each expression for u = 4i + 7j and v = 2i + 5j. Solution a. u – 3v = (4i + 7j) – 3(2i + 5j) a. u – 3v b. ||u – 3v|| = 4i + 7j – 6i – 15j = (4 – 6)i + (7 – 15)j = –2i – 8j

27. Slide 7.3 - 27 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley VECTORS IN TERMS OF MAGNITUDE AND DIRECTION Let that  is the smallest positive angle that v makes with the positive x-axis. The angle  is called the direction angle of v. The formula be a position vector, and suppose expresses a vector v in terms of its magnitude ||v|| and its direction angle .

28. Slide 7.3 - 28 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Writing a Vector with Given Length and Direction Angle Solution Write the vector with magnitude 3 that makes an angle of with the positive x-axis.

29. Slide 7.3 - 29 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THE DOT PRODUCT For two vectors the dot product of v and w, denoted v w, is defined as:

30. Slide 7.3 - 30 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Finding the Dot Product Find the dot product v w. Solution

31. Slide 7.3 - 31 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PROPERTIES OF THE DOT PRODUCT If u, v, and w are vectors and c is a scalar, then

32. Slide 7.3 - 32 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THE DOT PRODUCT AND THE ANGLE BETWEEN TWO VECTORS Let  (0 ≤  ≤ π) be the angle between two nonzero vectors v and w. Then: or

33. Slide 7.3 - 33 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 11 Using Vectors in Air Navigation An F-15 fighter jet is flying over Mount Rushmore at an airspeed (speed in still air) of 800 miles per hour on a bearing of N 30º E. The velocity of wind is 40 miles per hour in the direction of S 45º E. Find the actual speed and direction (relative to the ground) of the plane. Round each answer to the nearest tenth. Set up a coordinate system with north along the positive y-axis. Solution

34. Slide 7.3 - 34 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 11 Using Vectors in Air Navigation Let v be the air velocity (plane), w be the wind velocity, and r be the resultant ground velocity of the plane. Then, Solution continued

35. Slide 7.3 - 35 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution continued EXAMPLE 11 Using Vectors in Air Navigation The ground speed of the F-15 is approximately 790.6 miles per hour. To find the actual direction (bearing) of the plane, find the angle  between r and j (unit vector in the direction north).

36. Slide 7.3 - 36 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution continued EXAMPLE 11 Using Vectors in Air Navigation The direction of the F-15 relative to the ground is approximately N 32.8º E.

37. Slide 7.3 - 37 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DEFINITION OF WORK The work W done by a constant force F in moving an object from a point P to a point Q is defined by

38. Slide 7.3 - 38 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 12 Computing Work A child pulls a wagon along a level ground, with a force of 40 pounds along the handle on the wagon that makes an angle of 42º with the horizontal. How much work has she done by pulling the wagon 150 feet? Solution