SECTION 9.5 The Dot Product 651 force, then Bill must pull with how much force in order for the boulder to move? Tyrone Bill Chuck 500 lb 308 258 103. Challenge Problem See Problem 102. If Bill pulls due east with 200 pounds of force, then what direction does the boulder move? 101. Static Equilibrium Show on the following graph the force needed for the object at P to be in static equilibrium. F1 F2 F3 F4 P 102. Challenge Problem Landscaping. To drag a 500-pound boulder into place,Tyrone, Bill, and Chuck attach three ropes to the boulder as shown in the diagram. If Tyrone pulls with 240 pounds of force and Chuck pulls with 110 pounds of Explaining Concepts 104. Explain in your own words what a vector is. Give an example of a vector. 105. Write a brief paragraph comparing the algebra of complex numbers and the algebra of vectors. 106. Explain the difference between an algebraic vector and a position vector. Problems 107–116 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. Retain Your Knowledge 107. Solve: − = x 2 3 3 108. Factor − + + x x x 3 12 36 3 2 completely. 109. Find the exact value of ( ) ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ − tan cos 1 2 . 1 110. Find the amplitude, period, and phase shift of π ( ) = + y x 3 2 cos 6 3 Graph the function, showing at least two periods. 111. Find the distance between the points ( ) − − 5, 8 and ( ) 7, 1 . 112. Write the equation of the circle in standard form: + − + + = x y x y 20 4 55 0 2 2 113. Find all the intercepts of the graph of ( ) = + − − f x x x x 2 9 18 3 2 114. Solve: ( ) − + = x 4 5 9 53 2 115. If ( ) = f x x ,4 find ( ) ( ) − − f x f x 3 3 . 116. If θ θ ( ) = − f 25 2 and θ θ π θ π ( ) = − ≤ ≤ g 5sin , 2 2 , show that θ θ ( )( ) = f g 5 cos . OBJECTIVES 1 Find the Dot Product of Two Vectors (p. 651) 2 Find the Angle between Two Vectors (p. 652) 3 Determine Whether Two Vectors Are Parallel (p. 653) 4 Determine Whether Two Vectors Are Orthogonal (p. 654) 5 Decompose a Vector into Two Orthogonal Vectors (p. 654) 6 Compute Work (p. 656) 9.5 The Dot Product Now Work the ‘Are You Prepared?’ problem on page 657. • Law of Cosines (Section 8.3, pp. 570–573) PREPARING FOR THIS SECTION Before getting started, review the following: 1 Find the Dot Product of Two Vectors In Section 9.4, we defined the product of a scalar and a vector. Here we define the product of two vectors, called a dot product . We begin by defining the dot product algebraically. Later we will introduce a variety of geometric applications.
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