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Properties of the Dot Product Commutative Property: p ! # q ! q! # p ! , Distributive Property: Magnitudes Property: p ! # ! ! p 0 p 0 2 , p ! # 1q ! r ! 2 p! # q ! p! # r ! , Associative Property with a scalar K: 1kp ! 2 # q ! p! # 1kq ! 2 k1 p! # q ! 2 EXAMPLE 3 Selecting a strategy to determine the angle between two geometric vectors If the vectors and 4a ! b ! are perpendicular, and 0 a ! 0 2@ b ! @ , determine the angle (to the nearest degree) between the nonzero vectors a ! and b ! . a ! 3b ! CHAPTER 7 Solution Since the two given vectors are perpendicular, . Multiplying, (Distributive property) Simplifying, (Commutative property) Since 0 a ! 4 0 a ! 0 2 (Squaring both sides) Substituting, 414 @b ! 0 2@ b ! 11a ! ! ! # b 3@ b @ 2 @ , 0 a ! 0 2 @ 2 2 11112 @b ! 12@ @ 21@b ! b ! 0 @ 2 2 4@ b ! @ 2 @ 2cos u2 3 @b ! @ 2 0 Solving for cos u, cos u 13 @b! @ 2 4a ! # a ! a! # b ! 12b ! # a ! 3b ! # b ! 0 22 @b ! @ 2 cos u 13 22 , @ b! @ 2 0 a ! # 14a ! b ! 2 3b ! # 14a ! b ! 2 0 1a ! 3b ! 2 # 14a ! b ! 2 0 Thus, cos 1 a 13 b u, u 126.2° 22 Therefore, the angle between the two vectors is approximately 126.2°. It is often necessary to square the magnitude of a vector expression. This is illustrated in the following example. EXAMPLE 4 Proving that two vectors are perpendicular using the dot product If 0 x ! y ! 0 0 x ! y ! 0 , prove that the nonzero vectors, x ! and y ! , are perpendicular. NEL 375
Solution Consider the following diagram. x x + y –y x – y y Since 0x ! y ! 0 0x ! y ! 0 , x 0 x ! y ! 0 2 0 x ! y ! 0 2 ! ! and Therefore, 1x ! 0 x ! y ! y ! 0 2 1x ! y ! 2 # ! ! 0x y 0 2 1x ! y ! 2 # ! ! 1x y 2 1x y 2 ! ! 2 # ! ! 1x y 2 # ! ! 1x y 2 1x y 2 0 x ! 0 2 2x ! # y ! 0 y ! 0 2 0 x ! 0 2 2x ! # y ! 0 y ! 0 2 (Squaring both sides) (Magnitudes property) (Multiplying out) So, 4x ! # ! y 0 and x! # ! y 0 Thus, the two required vectors are shown to be perpendicular. (Geometrically, this means that if diagonals in a parallelogram are equal in length, then the sides must be perpendicular. In actuality, the parallelogram is a rectangle.) In this section, we dealt with the dot product and its geometric properties. In the next section, we will illustrate these same ideas with algebraic vectors. IN SUMMARY Key Idea • The dot product between two geometric vectors and is a scalar quantity defined as a ! ! # ! ! a ! b ! b 0a 0 @ b @ cos u , where u is the angle between the two vectors. Need to Know • If , then • If , then a ! a ! ! 0° u 6 90° # b 7 0 ! u 90° # b 0 SUMMARY • If , then • a ! ! # ! a ! ! 90° 6 u 180° # b 6 0 b b # ! a • a ! ! # ! ! 1b c 2 a! # b a! # ! c • a ! # ! ! a 0a 0 • ! ! ! 2 ! and k ! ! i # i 1, j # j 1, # k 1 • 1ka ! 2 # b ! a! # 1kb ! 2 k1a! # b ! 2 376 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
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R 2 EXAMPLE 1 Representing vectors
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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
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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 and 372: A T C 9. A small airplane has an ai
Page 373 and 374: Perhaps the most important observat
Page 375: If we look at the right-angled tria
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
Page 423 and 424: Chapter 7 Test 1. Given the vectors
Page 425 and 426: Review of Prerequisite Skills In th
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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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Page 481 and 482:
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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Review of Prerequisite Skills In th
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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
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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
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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
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Review Exercise 1. The lines 2x y
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14. You are given the lines , and r
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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
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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.
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14 Graphing a Trigonometric Functio
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2. Enter the second equation. Press
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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
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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
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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.
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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,
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Section 6.7, pp. 332-333 1. a. 1i !
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24. 25. a. 0a ! b. 0a ! 0 2 0b !
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. Since the equilibrant is directed
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7. a. scalar projection: 0, vector
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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.
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8. a. b. y 1 z 1 4 , 2 c. x 3t
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10. vector equation: r ! 10, 0, 42
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. 23. a. b. 2a 1 2 b a 6 7 , 2 7 ,
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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
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