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MHR • Calculus and Vectors 12 Solutions 679 Chapter 7 Cartesian Chapter 7 Cartesian Vectors Chapter 7 Prerequisite Skills Chapter 7 Prerequisite Skills Question 1 Page 358 Chapter 7 Prerequisite Skills Question 2 Page 358 Chapter 7 Prerequisite Skills Question 3 Page 358 a) 18° MHR • Calculus and Vectors 12 Solutions 679 b) 282° c) 0° d) 270° e) 240° f) 150° MHR • Calculus and Vectors 12 Solutions 680 g) 90° h) 70° i) 285° j) 220° k) 170° MHR • Calculus and Vectors 12 Solutions 681 l) 260° Chapter 7 Prerequisite Skills Question 4 Page 358 a) (3! 5)2 + (1! 6)2 = 4 + 25 = 29 The distance is 29 units. b) (6 ! (!4))2 + (7 ! 3)2 = 102 + 42 = 116 = 2 29 The distance is 2 29 units. c) (!5! (!1))2 + (8 ! 0)2 = (!4)2 + 82 = 80 = 4 5 The distance is 4 5 units. d) (!3! 5)2 + (!9 ! (!2))2 = (!8)2 + (!7)2 = 113 The distance is 113 units. MHR • Calculus and Vectors 12 Solutions 682 Chapter 7 Prerequisite Skills Question 5 Page 358 Answers may vary. For example: Let a = 4, x = 3, y = 7, and z = 5. a) Verify: x + y = y + x. L.S. = x + y = 3 + 7 = 10 R.S. = y + x = 7 + 3 = 10 Therefore, L.S. = R.S. In words, when adding two numbers, the order of the operation does not matter. b) Verify: x × y = y × x. L.S. = x × y = 3 × 7 = 21 R.S. = y × x = 7 × 3 = 21 Therefore, L.S. = R.S. In words, when multiplying two numbers, the order of the operation does not matter. c) Verify: (x + y) + z = x + (y + z). L.S. = (x + y) + z = (3 + 7) + 5 = 10 + 5 = 15 R.S. = x + (y + z) = 7 + (3 + 5) = 7 + 8 = 15 Therefore, L.S. = R.S. In words, when adding three numbers at a time, the grouping of the operations does not matter. MHR • Calculus and Vectors 12 Solutions 683 d) Verify: (x × y) × z = x × (y × z). L.S. = (x × y) × z = (3 × 7) × 5 = 21 × 5 = 105 R.S. = x × (y × z) = 7 × (3 × 5) = 7 × 15 = 105 Therefore, L.S. = R.S. In words, when multiplying three numbers at a time, the grouping of the operations does not matter. e) Verify: a(x + y) = ax + ay. L.S. = a(x + y) = 4(3 + 7) = 4(10) = 40 R.S. = ax + ay = 4(3) + 4(7) = 12 + 28 = 40 Therefore, L.S. = R.S. In words, when multiplying a binomial by a factor, you can multiply each of the terms in the binomial separately and then add the partial products together. f) Verify: x – y ≠ y – x. L.S. = x – y = 3 – 7 = –4 R.S. = y – x = 7 – 3 = 4 Therefore, L.S. ≠ R.S. In words, when subtracting two numbers, the order of the operation does matter. When the order of operation is reversed the answer is of the opposite sign. Chapter 7 Prerequisite Skills Question 6 Page 358 a) 5x + 3y = 11 2x + y = 4 –5x – 3y = –11 –1 6x + 3y = 12 3 x = 1 –1 + 3 MHR • Calculus and Vectors 12 Solutions 684 Substitute x = 1 into equation . 2(1) + y = 4 y = 2 Therefore, (x, y) = (1, 2). b) 2x + 6y = 14 x – 4y = –14 2x + 6y = 14 –2x + 8y = 28 –2 14y = 42 – 2 y = 3 Substitute y = 3 into equation . x ! 4(3) = !14 x = !2 Therefore, (x, y) = (–2, 3). c) 3x – 5y = –5 –6x + 2y = 2 6x – 10y = –10 2 –6x + 2y = 2 –8y = –8 2 + y = 1 Substitute y = 1 into equation . !6x + 2(1) = 2 !6x = 0 x = 0 Therefore, (x, y) = (0, 1). d) –1.5x + 3.2y = 10 0.5x + 0.4y = 4 –1.5x + 3.2y = 10 1.5x + 1.2y = 12 3 4.4y = 22 + 3 y = 5 Substitute y = 5 into equation . 0.5x + 0.4(5) = 4 0.5x = 2 x = 4 Therefore, (x, y) = (4, 5). MHR • Calculus and Vectors 12 Solutions 685 Chapter 7 Prerequisite Skills Question 7 Page 358 B is the correct response. It is not equivalent. 1 A The equation can be multiplied by to get this result. 3 B If you rearrange the equation into slope y-intercept form, the result is 9x ! 6y =18 !6y = !9x +18 !9 18 y = x + !6 !6 3 y = x ! 3 2 This is not the same as B. 1 C The equation can be multiplied by ! and rearranged to get this result. 3 1 D The equation can be multiplied by to get this result. 9 9x ! 6y =18 9 6 18 x ! y = 9 9 9 2 x ! y = 2 3 Chapter 7 Prerequisite Skills Question 8 Page 359 Use the special triangles. 1 3 1 a) b) c) d) 2 2 2 e) f) 0 g) 1 h) 0 i) cos 120º = –cos 60º j) sin 300º = –sin 60º k) 1 l) –1 1 3 = ! = ! 2 2 MHR • Calculus and Vectors 12 Solutions 686 Chapter 7 Prerequisite Skills Question 9 Page 359 a) 0.3 b) 0.7 c) −0.6 d) −0.9 e) 0.8 f) 0.8 Chapter 7 Prerequisite Skills Question 10 Page 359 a) Use the Pythagorean theorem x2 + 52 = 132 x2 169 25 = ! x2 = 144 x = 12 Therefore, x = 12 cm. b) Use the Pythagorean theorem. y2 = 4.52 + 8.92 y2 = 20.25+ 79.21 y2 = 99.46 y ! 9.97 y 10.0 ! Therefore, y ! 10.0 cm. Chapter 7 Prerequisite Skills Question 11 Page 359 h a) sin38o = 5 h = 5sin38° b) 32 + 42 = 52 The missing side of the base triangle has a length of 4 units. 4 sin30! = h 4 h = ! sin30 Chapter 7 Prerequisite Skills Question 12 Page 359 2 a) (a1 + b1 ) = (a1 + b1 )(a1 + b1 ) 2 2 = a1 + a1b1 + a1b1 + b1 = a 2 + 2a b + b 2 1 1 1 1 MHR • Calculus and Vectors 12 Solutions 687 2 2 b) (a1 + b1 )(a1 ! b1 ) = a1 ! a1b1 + a1b1 ! b1 2 2 = a1 ! b1 2 2 2 2 2 2 2 2 2 2 2 2 c) (a1 + a2 )(b1 + b2 ) = a1 b1 + a1 b2 + a2 b1 + a2 b2 3 d) (a1b1 + a2b2 + a3b3 ) = [(a1b1 + a2b2 + a3b3 )(a1b1 + a2b2 + a3b3 )](a1b1 + a2b2 + a3b3 ) 2 2 2 2 2 2 = [a1 b1 + a1b1a2b2 + a1b1a3b3 + a2b2a1b1 + a2 b2 + a2b2a3b3 + a3b3a1b1 + a3b3a2b2 + a3 b3 ](a1b1 + a2b2 + a3b3 ) 3 3 2 2 2 2 2 2 2 2 2 2 2 2 = a1 b1 + a1 a2b1 b2 + a1 a3b1 b3 + a1 a2b1 b2 + a1a2 b1b2 + a1a2a3b1b2b3 + a1 a3b1 b3 + a1a2a3b1b2b3 + a1a3 b1b3 2 2 2 2 2 2 3 3 2 2 2 2 2 2 + a1 a2b1 b2 + a1a2 b1b2 + a1a2a3b1b2b3 + a1a2 b1b2 + a2 b2 + a2 a3b2 b3 + a1a2a3b1b2b3 + a2 a3b2 b3 + a2a3 b2b3 2 2 2 2 2 2 2 2 2 2 2 2 3 3 + a1 a3b1 b3 + a1a2a3b1b2b3 + a1a3 b1b3 + a1a2a3b1b2b3 + a2 a3b2 b3 + a2a3 b2b3 + a1a3 b1b3 + a2a3 b2b3 + a3 b3 = a 3b 3 + a 3b 3 + a 3b 3 + 3a 2a b 2b + 3a a 2b b 2 + 3a 2a b 2b + 3a a 2b b 2 + 3a 2a b 2b + 3a a 2b b 2 + 6a a a b b b 1 1 2 2 3 3 1 2 1 2 1 2 1 2 1 3 1 3 1 3 1 3 2 3 2 3 2 3 2 3 1 2 3 1 2 3 Chapter 7 Prerequisite Skills Question 13 Page 359 1 a) A = 0(2) ! 0(0) + 0(2) ! 2(2) + 2(1) ! 3(2) + 3(!1) ! 3(1) + 3(0) ! 0(0) 2 1 = !14 2 = 7 The area is 7 square units. 1 b) A = 1(!2) ! 3(1) + 3(!4) ! 0(!2) + 0(0) ! (!5)(!4) + (!5)(4) ! (!3)(0) + (!3)(1) !1(4) 2 1 = !64 2 = 32 The area is 32 square units. 1 c) A = 4(!7) ! 5(!10) + 5(2) ! 3(!7) + 3(5) ! 0(2) + 0(3) ! (!2)(5) + (!2)(!10) ! 4(3) 2 1 = 86 2 = 43 The area is 43 square units. MHR • Calculus and Vectors 12 Solutions 688 Chapter 7 Prerequisite Skills Question 13 Page 359 Yes. There are similar formulas for triangles and hexagons. In fact, there are similar formulas for any polygon that does not intersect itself. For a triangle with vertices (x1,y1), (x2,y2), and (x3,y3), the area is 1 A = x y ! x y + x y ! x y + x y ! x y 2 1 2 2 1 2 3 3 2 3 1 1 3 Answers may vary. For example, Consider a triangle with vertices (1, 1), (2, 7), and (–3, –4), the area is 1 A = 1(7) ! 2(1) + 2(!4) ! (!3)(7) + (!3)(1) !1(!4) 2 1 = 19 2 = 9.5 The area is 9.5 square units. For a hexagon with vertices (x1,y1), (x2,y2), (x3,y3), (x4,y4), (x5,y5), and (x6,y6), the area is 1 A = x y ! x y + x y ! x y + x y ! x y + x y ! x y + x y ! x y + x y ! x y 2 1 2 2 1 2 3 3 2 3 4 4 3 4 5 5 4 5 6 6 5 6 1 1 6 For a hexagon with vertices (0, 0), (3, 2), (2, 4), (0, 5), (–2, 4), and (–3, 2), the area is 1 A = 0(2) ! 3(0) + 3(4) ! 2(2) + 2(5) ! 0(4) + 0(4) ! (–2)(5) + (–2)(2) ! (!3)(4) + (!3)(0) ! 0(2) 2 1 = 36 2 = 18 The area is 18 square units.
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