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Section 7.3—The Dot Product of Two Geometric Vectors In Chapter 6, the concept of multiplying a vector by a scalar was discussed. In this section, we introduce the dot product of two vectors and deal specifically with geometric vectors. When we refer to geometric vectors, we are referring to vectors that do not have a coordinate system associated with them. The dot product for any two vectors is defined as the product of their magnitudes multiplied by the cosine of the angle between the two vectors when the two vectors are placed in a tail-to-tail position. Dot Product of Two Vectors C A u AC ! # AB ! @AC ! @@AB ! @ cos u, 0 u 180° B Observations about the Dot Product There are some elementary but important observations that can be made about this calculation. First, the result of the dot product is always a scalar. Each of the quantities on the right side of the formula above is a scalar quantity, and so their product must be a scalar. For this reason, the dot product is also known as the scalar product. Second, the dot product can be positive, zero, or negative, depending upon the size of the angle between the two vectors. Sign of the Dot Product ! For the vectors and b ! , a ! ! # ! ! a b 0 a 0 @ b @ cos u, 0 u 180° : • for so a ! ! 0 u 6 90°, cos u 7 0, # b 7 0 • for so a ! ! u 90°, cos u 0, # b 0 • for so a ! ! 90° 6 u 180°, cos u 6 0, # b 6 0 The dot product is only calculated for vectors when the angle u between the vectors is to 0° to 180° , inclusive. (For convenience in calculating, the angle between the vectors is usually expressed in degrees, but radian measure is also correct.) NEL CHAPTER 7 371
Perhaps the most important observation to be made about the dot product is that when two nonzero vectors are perpendicular, their dot product is always 0. This will have many important applications in Chapter 8, when we discuss lines and planes. EXAMPLE 1 Calculating the dot product of two geometric vectors Two vectors, a ! and b ! , are placed tail to tail and have magnitudes 3 and 5, respectively. There is an angle of between the vectors. Calculate a ! ! 120° # b . Solution Since 0 a ! 0 3, @ b ! @ 5, and cos 120° 0.5, a ! # b ! 13215210.52 7.5 Notice that, in this example, it is stated that the vectors are tail to tail when taking the dot product. This is the convention that is always used, since this is the way of defining the angle between any two vectors. INVESTIGATION A.Given vectors a ! and b ! where , and the angle between the vectors is calculate a ! b ! 0a ! 0 5 b ! 8 60°, # . B. For the vectors given in part A calculate b ! # a ! . What do you notice? a Will this relationship always hold regardless of the two vectors used and the 60° measure of the angle between them? Explain. C. For the vectors given in part A calculate a ! # a ! and . Based on your observations, what can you conclude about u ! u ! b ! # b ! # for any vector u ! ? D.Using the vectors given in part A and a third vector c ! , 0c ! a 0 4, as shown in the diagram, calculate each of the following without rounding: ! ! i. iii. a ! ! # ! 60° b c 1b c 2 20° ii. the angle between iv. b ! c ! and a ! a ! ! b # b a! # ! c c . E. Compare your results from part iii and iv in part D. What property does this demonstrate? Write an equivalent expression for c ! # ! ! 1a b 2 and confirm it a using the appropriate calculations with the vectors given in part D. F. Using the 3 vectors given above, explain why . G.A fourth vector d ! , d ! 1a ! ! # b 2 # ! c a! # ! 1b # ! 60° c 2 d 20° b 3, is given as shown in the diagram. Explain why c a ! # d ! a! # 1d ! 2 1a ! 2 # d ! 1a ! 2 # 1d ! 2 H.Using the vectors given, calculate , and c ! ! a ! # 0 ! b ! # 0 ! # 0 . What does this imply? 372 7.3 THE DOT PRODUCT OF TWO GEOMETRIC VECTORS NEL
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Chapter 1 INTRODUCTION TO CALCULUS
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y x 3 1 , if 3 6 t 6 0 t 6. A func
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What Is Calculus? Two simple geomet
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If a and b were radicals, squaring
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Exercise 1.1 PART A 1. Write the co
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Consider a curve y f 1x2 and a poi
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EXAMPLE 2 Tech Support For help gra
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The slope of the tangent at 13, 42
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INVESTIGATION 4 Tech Support For he
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4. Simplify each of the following d
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18. Copy the following figures. Dra
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A. Determine the average velocity o
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Instantaneous Velocity The velocity
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Solution a. C11002 10V100 1000 1
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Exercise 1.3 C PART A 1. The veloci
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f 1x2 f 1a2 15. Use the alternate
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. Use your results for part a to ap
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The graph shown on your calculator
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limits because, in each case, only
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11. For each function, sketch the g
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EXAMPLE 2 EXAMPLE 3 EXAMPLE 4 Using
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INVESTIGATION Here is an alternate
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In this section, we learned the pro
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A 11. Jacques Charles (1746-1823) d
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A second condition for continuity a
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d. The function F is not continuous
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x 3, if x 3 12. g1x2 e 2 k, if
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Key Concepts Review We began our in
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c. What is the present rate of chan
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16. Evaluate the limit of each diff
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Chapter 2 DERIVATIVES Imagine a dri
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2 4 5. Expand, and collect like te
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Section 2.1—The Derivative Functi
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For the Parabola f1x2 x 2 The slop
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Note that f 1t2 Vt is defined for
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EXAMPLE 6 Reasoning about different
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Exercise 2.1 PART A 1. State the do
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T 15. Match each function in graphs
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EXAMPLE 1 Applying the constant fun
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We conclude this section with the s
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EXAMPLE 6 Connecting the derivative
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x 1 8. Determine the slope of the
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Section 2.3—The Product Rule In t
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EXAMPLE 3 Selecting an efficient st
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EXAMPLE 6 Selecting a strategy to d
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PART B dy 5. Determine the value of
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9. Determine the equation of the ta
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EXAMPLE 1 Applying the quotient rul
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Exercise 2.4 PART A 1. What are the
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Section 2.5—The Derivatives of Co
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with respect to x is equal to the p
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EXAMPLE 7 Combining derivative rule
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Exercise 2.5 PART A 1. Given f 1x2
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Technology Extension: Derivatives o
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Key Concepts Review Now that you ha
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9. For what values of x does the cu
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24. At a manufacturing plant, produ
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Chapter 3 DERIVATIVES AND THEIR APP
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4. Determine the area of each figur
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Section 3.1—Higher-Order Derivati
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Motion on a Straight Line An object
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d. The object moves in a positive d
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EXAMPLE 5 Analyzing motion under gr
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Exercise 3.1 C K PART A 1. Explain
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T 12. A dragster races down a 400 m
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H. Repeat part D for the following
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Only t 3 is in the given interval,
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IN SUMMARY Key Ideas • The maximu
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T PART B 4. Using the algorithm for
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Mid-Chapter Review 1. Determine the
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Section 3.3—Optimization Problems
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For extreme values, set V¿1x2 0.
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Exercise 3.3 C PART A 1. A piece of
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a. Determine the dimensions that sh
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The revenue function is R1x2 17 0
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The minimal cost is approximately $
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T C 8. The fuel cost per hour for r
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CAREER LINK WRAP-UP Investigate and
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10. For each of the following cost
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s1t2 t 26. Find the absolute maxi
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Chapter 4 CURVE SKETCHING If you ar
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5. Determine the derivative of each
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Section 4.1—Increasing and Decrea
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dy Since this is a polynomial funct
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IN SUMMARY Key Ideas • A function
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9. Each of the following graphs rep
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Solution dy First determine dx . dy
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Note that there is no value of x fo
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EXAMPLE 4 Graphing the derivative g
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4. Find the x- and y-intercepts of
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Section 4.3—Vertical and Horizont
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The behaviour of the graph can be i
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EXAMPLE 2 Expressing a polynomial f
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When lim f 1x2 k or lim f 1x2 k,
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For horizontal asymptotes, 3x f 1x2
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Now let's consider the straight lin
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Exercise 4.3 PART A 1. State the eq
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12. Use the features of each functi
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11. a. What does f ¿1x2 7 0 imply
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From these investigations, we can m
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Setting dy dx 0, we obtain 3 1x 2
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The second derivative of f 1x2 is f
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Exercise 4.4 K PART A 1. For each f
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Section 4.5—An Algorithm for Curv
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Therefore, by the second derivative
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Interval 1, 12 11, 0.82 10.8, 22 12
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K A 1x PART B 4. Use the algorithm
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Key Concepts Review In this chapter
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5. For each of the following, check
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17. If f 1x2 x3 8 , determine the
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Chapter 5 DERIVATIVES OF EXPONENTIA
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Radian Measure A radian is the meas
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5. Convert the following angles to
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Section 5.1—Derivatives of Expone
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From the investigation, you should
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EXAMPLE 4 Connecting the derivative
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A C PART B 6. Determine the equatio
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Section 5.2—The Derivative of the
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In general, for the exponential fun
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Solution a. January 1, 1900, is exa
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Section 5.3—Optimization Problems
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The average revenue to the company
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Exercise 5.3 PART A 1. Use graphing
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12. Find the maximum and minimum va
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9. The rapid growth in the number o
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INVESTIGATION 2 A.Using your graphi
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With practice, you will learn how t
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We evaluate f 1x2 at the critical n
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A T 6. Determine the absolute extre
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Derivatives of Composite Functions
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Review Exercise 1. Differentiate ea
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18. The position of a particle is g
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Chapter 5 Test dy 1. Determine the
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8. Use algebraic methods to evaluat
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f 1x2 142e 5x1 24. For each functi
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Review of Prerequisite Skills In th
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CAREER LINK Investigate CHAPTER 6:
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What is a Vector? A vector is a mat
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EXAMPLE 1 B A E D C Connecting vect
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A K B 5. Given the vector AB ! as s
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Section 6.2—Vector Addition In th
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the overall magnitude would be equa
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Solution c b a a - b + c a - b c a
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W N S E The vectors are drawn so th
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Exercise 6.2 PART A 1. The vectors
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10. a. In ! the ! example ! involvi
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0 0 The previous example illustrate
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Solution a. 30˚ y x 2x-y 2x -y 30
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Solution Draw a sketch and determin
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8. The two vectors a ! and b ! are
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Section 6.4—Properties of Vectors
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Demonstrating the distributive law
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IN SUMMARY Key Idea • Properties
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Mid-Chapter Review 1. ABCD is a par
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Section 6.5—Vectors in R 2 and R
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Right-Handed System of Coordinates
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Solution a. A16, 0, 02 is a point o
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There is one further observation th
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C A 14. a. What is the equation of
Page 321 and 322: R 2 EXAMPLE 1 Representing vectors
Page 323 and 324: EXAMPLE 2 Representing the sum and
Page 325 and 326: EXAMPLE 5 Calculating the magnitude
Page 327 and 328: A 10. Parallelogram is determined b
Page 329 and 330: EXAMPLE 1 Representing vectors in R
Page 331 and 332: . Using components, a ! b ! c !
Page 333 and 334: IN SUMMARY Key Ideas • In R 3 , i
Page 335 and 336: Section 6.8—Linear Combinations a
Page 337 and 338: In R 2 , it is possible to take any
Page 339 and 340: Geometrically, the linear combinati
Page 341 and 342: IN SUMMARY Key Ideas • In R 2 , O
Page 343 and 344: CAREER LINK WRAP-UP Investigate and
Page 345 and 346: Review Exercise 1. Determine whethe
Page 347 and 348: 15. A rectangular prism measuring 3
Page 349 and 350: Chapter 6 Test 1. The vectors a ! ,
Page 351 and 352: Review of Prerequisite Skills In th
Page 353 and 354: Section 7.1—Vectors as Forces The
Page 355 and 356: of 6 cm, also pointing east. The ve
Page 357 and 358: is drawn approximately to scale, an
Page 359 and 360: The magnitude of the vertical compo
Page 361 and 362: ! T1 sin 105° 1961sin 45°2 T ! 1
Page 363 and 364: 49 34 30 cos CBA 1 cos CBA 2 120
Page 365 and 366: C T f 3 40 N O 35 N 45° 30 N f 2 2
Page 367 and 368: Solution We start by drawing positi
Page 369 and 370: v a v + w w 20 triangles, Solving t
Page 371: A T C 9. A small airplane has an ai
Page 375 and 376: If we look at the right-angled tria
Page 377 and 378: Solution Consider the following dia
Page 379 and 380: A T 10. What is the value of the do
Page 381 and 382: Expanding, we get b 2 1 2a 1 b 1
Page 383 and 384: . 0 a ! 0 2 @b ! 112 2 122 2 142
Page 385 and 386: One of the most important propertie
Page 387 and 388: 4. a. From the set of vectors e11,
Page 389 and 390: Mid-Chapter Review 1. a. If 0 a ! 0
Page 391 and 392: Section 7.5—Scalar and Vector Pro
Page 393 and 394: EXAMPLE 1 Reasoning about the chara
Page 395 and 396: x EXAMPLE 3 z P (2, 1, 4) g b a O(0
Page 397 and 398: EXAMPLE 4 Calculating a specific di
Page 399 and 400: IN SUMMARY Key Idea • A projectio
Page 401 and 402: B C 12. In the diagram shown, ^ ABC
Page 403 and 404: a 3 b, k = 1 O b 3 a, k = -1 b a is
Page 405 and 406: If we carry out an identical proced
Page 407 and 408: EXAMPLE 2 Calculating cross product
Page 409 and 410: K A C T 4. Calculate the cross prod
Page 411 and 412: EXAMPLE 1 Using the dot product to
Page 413 and 414: . We start by constructing position
Page 415 and 416: EXAMPLE 4 Using the cross product t
Page 417 and 418: CAREER LINK WRAP-UP Investigate and
Page 419 and 420: Review Exercise 1. Given that a !
Page 421 and 422: 18. For the vectors m ! 1 V3, 2, 3
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Chapter 7 Test 1. Given the vectors
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Both of these vectors can be multip
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c. If the point Q121, 232 lies on t
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In this section, the vector and par
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C K A T 6. a. If the equation of a
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EXAMPLE 1 Representing the Cartesia
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It is not possible, in this case, t
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AP ! 1x 4, y 1222 1x 4, y 22.
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u cos 1 a 3 12 1 10 21 25 2 b u
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A T 10. For each pair of lines, det
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is its direction vector. If represe
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EXAMPLE 4 Representing the equation
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C A T 6. a. Determine parametric eq
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13. The Cartesian equation of a lin
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s t s(1, 2, 1) t(0, 2, 1), s, tR P
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After deriving vector and parametri
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EXAMPLE 4 Representing the equation
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A K T 7. a. Determine parameters co
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To derive the equation of this plan
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A number of observations can be mad
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When we considered lines in R 2 , w
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IN SUMMARY Key Idea • The Cartesi
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Section 8.6—Sketching Planes in R
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EXAMPLE 2 Representing the graph of
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EXAMPLE 5 Graphing planes whose Car
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IN SUMMARY Key Idea • A sketch of
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Review Exercise 1. Determine vector
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20. Calculate the acute angle that
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Chapter 8 Test 1. a. Given the poin
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4 13 s2 211 4s2 12 8s2 8 0 1
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Method 2: Again, this result can be
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We can now select any two of the th
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IN SUMMARY Key Ideas • Line and p
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A T 11. a. Show that the lines and
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equations. Since the lines would be
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Solution 1: Interchange equations 1
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Substituting into the second equati
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1: Multiply equation 1 by k, and ad
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K C PART B 4. Solve each system of
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Section 9.3—The Intersection of T
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If we had solved the system using e
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then x 3s 1. Now it is a matter o
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Exercise 9.3 C K PART A 1. A system
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Mid-Chapter Review 1. Determine the
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Section 9.4—The Intersection of T
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1: Create two new equations, 4 and
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Thus, 7y 5t 3. Dividing by 7, we g
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Inconsistent Systems for Three Equa
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Therefore, we can choose ! ! ! m 1
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Exercise 9.4 PART A 1. A student is
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9. Solve each system of equations u
Page 535 and 536:
Section 9.5—The Distance from a P
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EXAMPLE 1 Calculating the distance
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In triangle PQR, sin u d @ QP ! @
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Exercise 9.5 PART A 1. Determine th
Page 543 and 544:
Section 9.6—The Distance from a P
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It is also possible to use the dist
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p 1 p 2 U V d L 1 : r = (-2, 1, 0)
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The required distance between the t
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K A T PART B 2. Determine the follo
Page 553 and 554:
Review Exercise 1. The lines 2x y
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14. You are given the lines , and r
Page 557 and 558:
Chapter 9 Test 1. a. Determine the
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P(1, -2, 4) Q P9 2x - 3y - 4z + 66
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29. A line that passes through the
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Solution a. Differentiate both side
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Exercise PART A 1. State the chain
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dA b. To determine differentiate A
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1 16 dh dt Therefore, at the momen
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11. Two cyclists depart at the same
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Solution a. y ln15x2 Using the cha
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x1x 42 0 x 4 or x 0 But x 0 is
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The Derivatives of General Logarith
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The Derivative of a Composite Funct
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To solve this, we take the natural
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Exercise PART A 1. Differentiate ea
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y eliminating unknowns. In Example
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Properties of a Matrix in Row-Echel
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Exercise PART A 1. Write an augment
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12. Determine the equation of the p
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1 0 0 18 4 (row 3) row 1 £ 0 1 0
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Finally, convert the entry in the s
Page 597 and 598:
Review of Technical Skills Appendix
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3 Evaluating a Function 1. Enter th
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7 Making a Table of Differences To
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4. Cursor along the curve to any po
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5. Display and analyze the results.
Page 607 and 608:
14 Graphing a Trigonometric Functio
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2. Enter the second equation. Press
Page 611 and 612:
Use either the calculator keypad or
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20 Graphing the Derivative of a Fun
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4. Create a function. Right-click t
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composition: the process of combini
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G geometric vectors: vectors that a
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ationalizing the denominator: the p
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Answers Chapter 1 0 0 0 0 0 Review
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d. about 1 e. about 7 8 f. no tang
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d. 13. m 3; b 1 14. a 3, b 2, c
Page 629 and 630:
Review Exercise, pp. 56-59 1. a. 3
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19. a. y 7 b. y 5x 5 c. y 18x 9
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20. a. 34.3 m> s b. 39.2 m> s c. 54
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17. a. 100 bacteria b. 1200 bacteri
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11. a. b. 160x y 16 0 60x y 61
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3. 1 2x 4. a. x 2 15x 6 b. 6012x
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14. t 1 s; away 15. a. s1t2 kt 2
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14. a. triangle side length 0.96 cm
Page 645 and 646:
Review Exercise, pp. 156-159 11. 20
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c. i. ii. iii. d. i. ii. iii. 5 4 3
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. local minimum: 11.41, 39.62, loca
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2. increasing: x 6 1 and x 7 2, dec
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to a zero of its derivative, the nu
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7. (-2, 10) 10 8 y 10. a. 8 y e. 4
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4. hole at x 2; large and negative
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sin x b. 1 sin 2 tan x sec x x L
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12. a. no maximum or minimum value
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Review Exercise, pp. 263-265 1. a.
Page 665 and 666:
x 2. 5. a. 19 000 fish> year b. 23
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Section 6.1, pp. 279-281 6. a. b. c
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17. P G 4. Answers may vary. For ex
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11. a. AG ! a ! b ! c ! , b. AG
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d. OD ! 11, 1, 12 (0, 0, 1) (1, 0,
Page 675 and 676:
Section 6.7, pp. 332-333 1. a. 1i !
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24. 25. a. 0a ! b. 0a ! 0 2 0b !
Page 679 and 680:
. Since the equilibrant is directed
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7. a. scalar projection: 0, vector
Page 683 and 684:
2. a. 3 b. 7 c. 43 d. 217 e. 5 f. 1
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cos A 1 cos B 1 cos C 1 area of t
Page 687 and 688:
. r ! 11, 1, 02 t10, 1, 12, tR; x
Page 689 and 690:
2. 12, 3, 42 3. P must lie on plane
Page 691 and 692:
25. A plane has two parameters, bec
Page 693 and 694:
coincident with the plane, meaning
Page 695 and 696:
Mid-Chapter Review, pp. 518-519 1.
Page 697 and 698:
8. a. b. y 1 z 1 4 , 2 c. x 3t
Page 699 and 700:
10. vector equation: r ! 10, 0, 42
Page 701 and 702:
. 23. a. b. 2a 1 2 b a 6 7 , 2 7 ,
Page 703 and 704:
16. 17. 18. 19. 2 3 4 m>min 144 m>m
Page 705 and 706:
ln x 1 e. ln 2 4x 1 x ln 13x 2 2
Page 707 and 708:
Index A Absolute extrema, 130, 132,
Page 709:
normal, 461-469, 486, 512, 524 para
show all

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