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### 7 vectors

1. 1. Vectors
2. 2. VectorsA vector, in 2 or higher dimensional space, is anumerical measurement in a specified direction.
3. 3. VectorsA vector, in 2 or higher dimensional space, is anumerical measurement in a specified direction.We use the symbols u, v, and w (or u, v, and w)to represent vectors in mathematics.
4. 4. VectorsA vector, in 2 or higher dimensional space, is anumerical measurement in a specified direction.We use the symbols u, v, and w (or u, v, and w)to represent vectors in mathematics. In physics,F, G, and H are used to represent force vectors.
5. 5. VectorsA vector, in 2 or higher dimensional space, is anumerical measurement in a specified direction.We use the symbols u, v, and w (or u, v, and w)to represent vectors in mathematics. In physics,F, G, and H are used to represent force vectors.We may draw an arrow to represent a vector,with the arrow pointing in the specified direction andthe length of the arrow indicating the numericalmeasurement. u
6. 6. VectorsA vector, in 2 or higher dimensional space, is anumerical measurement in a specified direction.We use the symbols u, v, and w (or u, v, and w)to represent vectors in mathematics. In physics,F, G, and H are used to represent force vectors.We may draw an arrow to represent a vector,with the arrow pointing in the specified direction andthe length of the arrow indicating the numericalmeasurement. We write the arrow or vector Bwith A as the base point and B asthe tip as AB . AB u A
7. 7. VectorsA vector, in 2 or higher dimensional space, is anumerical measurement in a specified direction.We use the symbols u, v, and w (or u, v, and w)to represent vectors in mathematics. In physics,F, G, and H are used to represent force vectors.We may draw an arrow to represent a vector,with the arrow pointing in the specified direction andthe length of the arrow indicating the numericalmeasurement. We write the arrow or vector Bwith A as the base point and B asthe tip as AB . Many vector related ABproblems may be solved using the ugeometric diagrams based on these Aarrows
8. 8. VectorsThe length of a vector is called the magnitude of u orthe absolute value of u. It is denoted as |u| and |u| ≥ 0.
9. 9. VectorsThe length of a vector is called the magnitude of u orthe absolute value of u. It is denoted as |u| and |u| ≥ 0. Vector Arithmetic
10. 10. VectorsThe length of a vector is called the magnitude of u orthe absolute value of u. It is denoted as |u| and |u| ≥ 0. Vector ArithmeticEquality of VectorsTwo vectors u and v with thesame length and same directionare equal, i.e. u = v.
11. 11. VectorsThe length of a vector is called the magnitude of u orthe absolute value of u. It is denoted as |u| and |u| ≥ 0. Vector ArithmeticEquality of Vectors vTwo vectors u and v with thesame length and same direction u u=vare equal, i.e. u = v.
12. 12. VectorsThe length of a vector is called the magnitude of u orthe absolute value of u. It is denoted as |u| and |u| ≥ 0. Vector ArithmeticEquality of Vectors vTwo vectors u and v with thesame length and same direction u u=vare equal, i.e. u = v.Scalar MultiplicationGiven a number λ and a vector v, λvis the extension or the compressionof the vector v by a factor λ.
13. 13. VectorsThe length of a vector is called the magnitude of u orthe absolute value of u. It is denoted as |u| and |u| ≥ 0. Vector ArithmeticEquality of Vectors vTwo vectors u and v with thesame length and same direction u u=vare equal, i.e. u = v.Scalar MultiplicationGiven a number λ and a vector v, λvis the extension or the compressionof the vector v by a factor λ. v 2v
14. 14. VectorsThe length of a vector is called the magnitude of u orthe absolute value of u. It is denoted as |u| and |u| ≥ 0. Vector ArithmeticEquality of Vectors vTwo vectors u and v with thesame length and same direction u u=vare equal, i.e. u = v.Scalar Multiplication –2v –vGiven a number λ and a vector v, λvis the extension or the compressionof the vector v by a factor λ. v 2vIf λ < 0, λv is in the oppositedirection of v.
15. 15. VectorsThe length of a vector is called the magnitude of u orthe absolute value of u. It is denoted as |u| and |u| ≥ 0. Vector ArithmeticEquality of Vectors vTwo vectors u and v with thesame length and same direction u u=vare equal, i.e. u = v.Scalar Multiplication –2v –vGiven a number λ and a vector v, λvis the extension or the compressionof the vector v by a factor λ. v 2vIf λ < 0, λv is in the oppositedirection of v. Finally, 0v = 0 , the zero vector.
16. 16. Vector AdditionVector Addition and The Parallelogram RuleGeometrically, given two vectors u and v,the vector u + v is the diagonal vector ofthe parallelogram as shown,
17. 17. Vector AdditionVector Addition and The Parallelogram RuleGeometrically, given two vectors u and v,the vector u + v is the diagonal vector ofthe parallelogram as shown, v u The Parallelogram Rule
18. 18. Vector AdditionVector Addition and The Parallelogram Rule uGeometrically, given two vectors u and v, vthe vector u + v is the diagonal vector of u+vthe parallelogram as shown, v u The Parallelogram Rule
19. 19. Vector AdditionVector Addition and The Parallelogram Rule uGeometrically, given two vectors u and v, vthe vector u + v is the diagonal vector of u+vthe parallelogram as shown, and u + v is v ualso called the resultant of u and v in The Parallelogram Rulephysics.
20. 20. Vector AdditionVector Addition and The Parallelogram Rule uGeometrically, given two vectors u and v, vthe vector u + v is the diagonal vector of u+vthe parallelogram as shown, and u + v is v ualso called the resultant of u and v in The Parallelogram Rulephysics.The Base to Tip RuleGiven two vectors u and v, u + v is thenew vector formed by placing the baseof one vector at the tip of the other.
21. 21. Vector AdditionVector Addition and The Parallelogram Rule uGeometrically, given two vectors u and v, vthe vector u + v is the diagonal vector of u+vthe parallelogram as shown, and u + v is v ualso called the resultant of u and v in The Parallelogram Rulephysics. uThe Base to Tip RuleGiven two vectors u and v, u + v is the vnew vector formed by placing the base The Base to Tip Ruleof one vector at the tip of the other.
22. 22. Vector AdditionVector Addition and The Parallelogram Rule uGeometrically, given two vectors u and v, vthe vector u + v is the diagonal vector of u+vthe parallelogram as shown, and u + v is v ualso called the resultant of u and v in The Parallelogram Rulephysics. uThe Base to Tip Rule vGiven two vectors u and v, u + v is the vnew vector formed by placing the base The Base to Tip Ruleof one vector at the tip of the other.
23. 23. Vector AdditionVector Addition and The Parallelogram Rule uGeometrically, given two vectors u and v, vthe vector u + v is the diagonal vector of u+vthe parallelogram as shown, and u + v is v ualso called the resultant of u and v in The Parallelogram Rulephysics. uThe Base to Tip Rule vGiven two vectors u and v, u + v is the v u+vnew vector formed by placing the base The Base to Tip Ruleof one vector at the tip of the other.
24. 24. Vector Addition Vector Addition and The Parallelogram Rule u Geometrically, given two vectors u and v, v the vector u + v is the diagonal vector of u+v the parallelogram as shown, and u + v is v u also called the resultant of u and v in The Parallelogram Rule physics. uThe Base to Tip Rule vGiven two vectors u and v, u + v is the v u+vnew vector formed by placing the base The Base to Tip Ruleof one vector at the tip of the other.It is easier to use the base-to-tip ruleto see that the order of the additiondoes not matter for the sum ofthree or more vectors.
25. 25. Vector Addition Vector Addition and The Parallelogram Rule u Geometrically, given two vectors u and v, v the vector u + v is the diagonal vector of u+v the parallelogram as shown, and u + v is v u also called the resultant of u and v in The Parallelogram Rule physics. uThe Base to Tip Rule vGiven two vectors u and v, u + v is the v u+vnew vector formed by placing the base The Base to Tip Ruleof one vector at the tip of the other.It is easier to use the base-to-tip rule v u wto see that the order of the additiondoes not matter for the sum ofthree or more vectors.
26. 26. Vector Addition Vector Addition and The Parallelogram Rule u Geometrically, given two vectors u and v, v the vector u + v is the diagonal vector of u+v the parallelogram as shown, and u + v is v u also called the resultant of u and v in The Parallelogram Rule physics. uThe Base to Tip Rule vGiven two vectors u and v, u + v is the v u+vnew vector formed by placing the base The Base to Tip Ruleof one vector at the tip of the other.It is easier to use the base-to-tip rule v u wto see that the order of the additiondoes not matter for the sum ofthree or more vectors. u
27. 27. Vector Addition Vector Addition and The Parallelogram Rule u Geometrically, given two vectors u and v, v the vector u + v is the diagonal vector of u+v the parallelogram as shown, and u + v is v u also called the resultant of u and v in The Parallelogram Rule physics. uThe Base to Tip Rule vGiven two vectors u and v, u + v is the v u+vnew vector formed by placing the base The Base to Tip Ruleof one vector at the tip of the other.It is easier to use the base-to-tip rule v u wto see that the order of the addition vdoes not matter for the sum ofthree or more vectors. u
28. 28. Vector Addition Vector Addition and The Parallelogram Rule u Geometrically, given two vectors u and v, v the vector u + v is the diagonal vector of u+v the parallelogram as shown, and u + v is v u also called the resultant of u and v in The Parallelogram Rule physics. uThe Base to Tip Rule vGiven two vectors u and v, u + v is the v u+vnew vector formed by placing the base The Base to Tip Ruleof one vector at the tip of the other.It is easier to use the base-to-tip rule v u wto see that the order of the addition v wdoes not matter for the sum ofthree or more vectors. u u+v+w
29. 29. Vector Addition Vector Addition and The Parallelogram Rule u Geometrically, given two vectors u and v, v the vector u + v is the diagonal vector of u+v the parallelogram as shown, and u + v is v u also called the resultant of u and v in The Parallelogram Rule physics. uThe Base to Tip Rule vGiven two vectors u and v, u + v is the v u+vnew vector formed by placing the base The Base to Tip Ruleof one vector at the tip of the other.It is easier to use the base-to-tip rule v u wto see that the order of the addition v wdoes not matter for the sum ofthree or more vectors. u
30. 30. Vector Addition Vector Addition and The Parallelogram Rule u Geometrically, given two vectors u and v, v the vector u + v is the diagonal vector of u+v the parallelogram as shown, and u + v is v u also called the resultant of u and v in The Parallelogram Rule physics. uThe Base to Tip Rule vGiven two vectors u and v, u + v is the v u+vnew vector formed by placing the base The Base to Tip Ruleof one vector at the tip of the other.It is easier to use the base-to-tip rule v u wto see that the order of the addition v wdoes not matter for the sum ofthree or more vectors. u u+v+w
31. 31. Vector Addition Vector Addition and The Parallelogram Rule u Geometrically, given two vectors u and v, v the vector u + v is the diagonal vector of u+v the parallelogram as shown, and u + v is v u also called the resultant of u and v in The Parallelogram Rule physics. uThe Base to Tip Rule vGiven two vectors u and v, u + v is the v u+vnew vector formed by placing the base The Base to Tip Ruleof one vector at the tip of the other.It is easier to use the base-to-tip rule v u wto see that the order of the addition v wdoes not matter for the sum ofthree or more vectors. u w u+v+w
32. 32. Vector Addition Vector Addition and The Parallelogram Rule u Geometrically, given two vectors u and v, v the vector u + v is the diagonal vector of u+v the parallelogram as shown, and u + v is v u also called the resultant of u and v in The Parallelogram Rule physics. uThe Base to Tip Rule vGiven two vectors u and v, u + v is the v u+vnew vector formed by placing the base The Base to Tip Ruleof one vector at the tip of the other.It is easier to use the base-to-tip rule v u wto see that the order of the addition v wdoes not matter for the sum ofthree or more vectors. u u w u+v+w
33. 33. Vector Addition Vector Addition and The Parallelogram Rule u Geometrically, given two vectors u and v, v the vector u + v is the diagonal vector of u+v the parallelogram as shown, and u + v is v u also called the resultant of u and v in The Parallelogram Rule physics. uThe Base to Tip Rule vGiven two vectors u and v, u + v is the v u+vnew vector formed by placing the base The Base to Tip Ruleof one vector at the tip of the other.It is easier to use the base-to-tip rule v u wto see that the order of the addition v wdoes not matter for the sum of vthree or more vectors. u u w u+v+w
34. 34. Vector Addition Vector Addition and The Parallelogram Rule u Geometrically, given two vectors u and v, v the vector u + v is the diagonal vector of u+v the parallelogram as shown, and u + v is v u also called the resultant of u and v in The Parallelogram Rule physics. uThe Base to Tip Rule vGiven two vectors u and v, u + v is the v u+vnew vector formed by placing the base The Base to Tip Ruleof one vector at the tip of the other.It is easier to use the base-to-tip rule v u wto see that the order of the addition v w+u+v wdoes not matter for the sum of = vthree or more vectors. u u w u+v+w
35. 35. Vector SubtractionThe Tip to Tip RuleGiven two vectors u and v, the vector vu – v = u +(–v) is the vector from the tip uof v to tip of u.
36. 36. Vector SubtractionThe Tip to Tip Rule u–vGiven two vectors u and v, the vector vu – v = u +(–v) is the vector from the tip uof v to tip of u.
37. 37. Vector SubtractionThe Tip to Tip Rule u–vGiven two vectors u and v, the vector vu – v = u +(–v) is the vector from the tip uof v to tip of u.Example A. Given the following three vectorsu, v and w, draw u – v – w. w u v
38. 38. Vector SubtractionThe Tip to Tip Rule u–vGiven two vectors u and v, the vector vu – v = u +(–v) is the vector from the tip uof v to tip of u.Example A. Given the following three vectorsu, v and w, draw u – v – w. w The vector u – v is u u v v
39. 39. Vector SubtractionThe Tip to Tip Rule u–vGiven two vectors u and v, the vector vu – v = u +(–v) is the vector from the tip uof v to tip of u.Example A. Given the following three vectorsu, v and w, draw u – v – w. u–v w The vector u – v is u u v v
40. 40. Vector SubtractionThe Tip to Tip Rule u–vGiven two vectors u and v, the vector vu – v = u +(–v) is the vector from the tip uof v to tip of u.Example A. Given the following three vectorsu, v and w, draw u – v – w. u–v w The vector u – v is u u v vHence the vector u – v – w is
41. 41. Vector SubtractionThe Tip to Tip Rule u–vGiven two vectors u and v, the vector vu – v = u +(–v) is the vector from the tip uof v to tip of u.Example A. Given the following three vectorsu, v and w, draw u – v – w. u–v w The vector u – v is u u v v u–vHence the vector u – v – w is
42. 42. Vector SubtractionThe Tip to Tip Rule u–vGiven two vectors u and v, the vector vu – v = u +(–v) is the vector from the tip uof v to tip of u.Example A. Given the following three vectorsu, v and w, draw u – v – w. u–v w The vector u – v is u u v v u–v wHence the vector u – v – w is
43. 43. Vector SubtractionThe Tip to Tip Rule u–vGiven two vectors u and v, the vector vu – v = u +(–v) is the vector from the tip uof v to tip of u.Example A. Given the following three vectorsu, v and w, draw u – v – w. u–v w The vector u – v is u u v v u–v wHence the vector u – v – w is u–v–w
44. 44. Vector SubtractionThe Tip to Tip Rule u–vGiven two vectors u and v, the vector vu – v = u +(–v) is the vector from the tip uof v to tip of u.Example A. Given the following three vectorsu, v and w, draw u – v – w. u–v w The vector u – v is u u v v u–v wHence the vector u – v – w is u–v–wYour turn: Draw u – (v + w). Show that it’s the sameas u – v – w.
45. 45. Vectors in a Coordinate SystemA vector v placed in the coordinatesystem with the base at the origin(0, 0) is said to be in the standardposition.
46. 46. Vectors in a Coordinate SystemA vector v placed in the coordinatesystem with the base at the origin(0, 0) is said to be in the standardposition. If the tip of the vector v inthe standard position is (a, b), we write v as <a, b>.
47. 47. Vectors in a Coordinate System y u = <3, 4>A vector v placed in the coordinatesystem with the base at the origin v = <–1, 2>(0, 0) is said to be in the standard xposition. If the tip of the vector v inthe standard position is (a, b), we write v as <a, b>.
48. 48. Vectors in a Coordinate System y u = <3, 4>A vector v placed in the coordinatesystem with the base at the origin v = <–1, 2>(0, 0) is said to be in the standard xposition. If the tip of the vector v inthe standard position is (a, b), we write v as <a, b>.The zero vector is 0 = <0, 0>.
49. 49. Vectors in a Coordinate System y u = <3, 4>A vector v placed in the coordinatesystem with the base at the origin v = <–1, 2>(0, 0) is said to be in the standard xposition. If the tip of the vector v inthe standard position is (a, b), we write v as <a, b>.The zero vector is 0 = <0, 0>. The vector BT,with the base point B = (a, b) and the tip T = (c, d), inthe standard form is BT = T – B= <c – a, d – b>.
50. 50. Vectors in a Coordinate System y u = <3, 4>A vector v placed in the coordinatesystem with the base at the origin v = <–1, 2>(0, 0) is said to be in the standard xposition. If the tip of the vector v inthe standard position is (a, b), we write v as <a, b>.The zero vector is 0 = <0, 0>. The vector BT,with the base point B = (a, b) and the tip T = (c, d), inthe standard form is BT = T – B= <c – a, d – b>.Hence the vector from T = (–2, 4)B = (3, –1), to T = (–2, 4) isBT = T – B = (–2, 4) – (3, –1) BT B = (3, –1)
51. 51. Vectors in a Coordinate System y u = <3, 4>A vector v placed in the coordinatesystem with the base at the origin v = <–1, 2>(0, 0) is said to be in the standard xposition. If the tip of the vector v inthe standard position is (a, b), we write v as <a, b>.The zero vector is 0 = <0, 0>. The vector BT,with the base point B = (a, b) and the tip T = (c, d), inthe standard form is BT = T – B= <c – a, d – b>.Hence the vector from T = (–2, 4)B = (3, –1), to T = (–2, 4) isBT = T – B = (–2, 4) – (3, –1) BT=<–2 – 3, 4 – (–1) >=<–5, 5 > B = (3, –1)in the standard form.
52. 52. Vectors in a Coordinate System y u = <3, 4>A vector v placed in the coordinatesystem with the base at the origin v = <–1, 2>(0, 0) is said to be in the standard xposition. If the tip of the vector v inthe standard position is (a, b), we write v as <a, b>.The zero vector is 0 = <0, 0>. The vector BT,with the base point B = (a, b) and the tip T = (c, d), inthe standard form is BT = T – B= <c – a, d – b>.Hence the vector from T = (–2, 4)B = (3, –1), to T = (–2, 4) isBT = T – B = (–2, 4) – (3, –1) BT = < –5, 5 > BT=<–2 – 3, 4 – (–1) >=<–5, 5 > BT in the standard B = (3, –1)in the standard form. form based at (0, 0)
53. 53. Magnitudes of VectorsThe magnitude or the length of v = <a, b> ythe vector v = <a, b> is given by |v| = √a2 + b2|v| = √a2 + b2 and the magnitude bof |BT| = √Δx2 + Δy2. a x
54. 54. Magnitudes of VectorsThe magnitude or the length of v = <a, b> ythe vector v = <a, b> is given by |v| = √a2 + b2|v| = √a2 + b2 and the magnitude bof |BT| = √Δx2 + Δy2. a xExample B.Find BT in the standard form |BT| if B = (1, –3) andT = (5, 1). Draw.
55. 55. Magnitudes of VectorsThe magnitude or the length of v = <a, b> ythe vector v = <a, b> is given by |v| = √a2 + b2|v| = √a2 + b2 and the magnitude bof |BT| = √Δx2 + Δy2. a xExample B.Find BT in the standard form |BT| if B = (1, –3) andT = (5, 1). Draw. y BT=<4, 4> T(5, 1) x B(1, –3)
56. 56. Magnitudes of VectorsThe magnitude or the length of v = <a, b> ythe vector v = <a, b> is given by |v| = √a2 + b2|v| = √a2 + b2 and the magnitude bof |BT| = √Δx2 + Δy2. a xExample B.Find BT in the standard form |BT| if B = (1, –3) andT = (5, 1). Draw. y BT=<4, 4>BT = T – B = (5, 1) – (1, –3) T(5, 1)= <4, 4> x B(1, –3)
57. 57. Magnitudes of VectorsThe magnitude or the length of v = <a, b> ythe vector v = <a, b> is given by |v| = √a2 + b2|v| = √a2 + b2 and the magnitude bof |BT| = √Δx2 + Δy2. a xExample B.Find BT in the standard form |BT| if B = (1, –3) and T = (5, 1). Draw. y BT=<4, 4>BT = T – B = (5, 1) – (1, –3) T(5, 1)= <4, 4> and that x|BT| = |<4, 4>| = √42 + 42 = √32 = 4√2 B(1, –3)
58. 58. Magnitudes of VectorsThe magnitude or the length of v = <a, b> ythe vector v = <a, b> is given by |v| = √a2 + b2|v| = √a2 + b2 and the magnitude bof |BT| = √Δx2 + Δy2. a xExample B.Find BT in the standard form |BT| if B = (1, –3) and T = (5, 1). Draw. y BT=<4, 4>BT = T – B = (5, 1) – (1, –3) T(5, 1)= <4, 4> and that x|BT| = |<4, 4>| = √42 + 42 = √32 = 4√2 B(1, –3)Given vectors in the standard form <a, b>,scalar-multiplication and vector addition are carriedout coordinate-wise, i.e. operations are done at eachcoordinate.
59. 59. Algebra of Standard VectorsScalar Multiplication yLet u = <a, b> and λ be a real number, xthen λu = λ<a, b> = <λa, λb>.Hence if u = <–2, 1> u=<–2, 1> 2u=2<–2, 1>then 2u = 2<–2, 1> = <–4, 2> =<–4, 2>(This corresponds to stretching u +v=<2, 4>u to twice its length.) u v v=<–2, 3>Vector Addition u+vLet u = <a, b>, v = <c, d> u=<4, 1> xthen u + v = <a + c, b + d>.This is the same as the Parallelogram Rule.Hence if u = <4, 1> and v = <–2, 3>then u + v = <4, 1> + <–2, 3> = <2, 4>.Finally, we define u – v = u + (–v)
60. 60. Algebra of Standard VectorsExample C. Let u = <–4, 1>, v = <3, –6>, finda. 5u – 3v
61. 61. Algebra of Standard VectorsExample C. Let u = <–4, 1>, v = <3, –6>, finda. 5u – 3v 5u – 3v = <–29, 23> u v
62. 62. Algebra of Standard VectorsExample C. Let u = <–4, 1>, v = <3, –6>, finda. 5u – 3v 5u – 3v = <–29, 23> 5u u v
63. 63. Algebra of Standard VectorsExample C. Let u = <–4, 1>, v = <3, –6>, finda. 5u – 3v 5u – 3v = <–29, 23> 5u –3v u v
64. 64. Algebra of Standard VectorsExample C. Let u = <–4, 1>, v = <3, –6>, finda. 5u – 3v 5u – 3v 5u –3v u v
65. 65. Algebra of Standard VectorsExample C. Let u = <–4, 1>, v = <3, –6>, finda. 5u – 3v 5u – 3v= 5<–4, 1> – 3<3, –6> 5u –3v u v
66. 66. Algebra of Standard VectorsExample C. Let u = <–4, 1>, v = <3, –6>, finda. 5u – 3v 5u – 3v = <–29, 23>= 5<–4, 1> – 3<3, –6>= <–20, 5> + <–9, 18>= <–29, 23> 5u –3v u v
67. 67. Algebra of Standard VectorsExample C. Let u = <–4, 1>, v = <3, –6>, finda. 5u – 3v 5u – 3v = <–29, 23>= 5<–4, 1> – 3<3, –6>= <–20, 5> + <–9, 18>= <–29, 23> 5u –3vb. |5u – 3v| u v
68. 68. Algebra of Standard VectorsExample C. Let u = <–4, 1>, v = <3, –6>, finda. 5u – 3v 5u – 3v = <–29, 23>= 5<–4, 1> – 3<3, –6>= <–20, 5> + <–9, 18>= <–29, 23> 5u –3vb. |5u – 3v|= √(–29)2 + (23)2 u= √1370 ≈ 37.0 v
69. 69. Algebra of Standard VectorsExample C. Let u = <–4, 1>, v = <3, –6>, finda. 5u – 3v 5u – 3v = <–29, 23>= 5<–4, 1> – 3<3, –6>= <–20, 5> + <–9, 18>= <–29, 23> 5u –3vb. |5u – 3v|= √(–29)2 + (23)2 u= √1370 ≈ 37.0 vYour TurnGiven the u and v above, find 3u – 5v and |3u –5v|.Sketch the vectors. 33> and |3u – 5v| ≈ 42.6Ans: 3u – 5v = <-27,
70. 70. Algebra of Standard VectorsExample C. Let u = <–4, 1>, v = <3, –6>, finda. 5u – 3v 5u – 3v = <–29, 23>= 5<–4, 1> – 3<3, –6>= <–20, 5> + <–9, 18>= <–29, 23> 5u –3vb. |5u – 3v|= √(–29)2 + (23)2 u= √1370 ≈ 37.0 vYour TurnGiven the u and v above, find 3u – 5v and |3u –5v|.Sketch the vectors. 33> and |3u – 5v| ≈ 42.6Ans: 3u – 5v = <–27,
71. 71. Algebra of Standard VectorsThe Unit Coordinate Vectors
72. 72. Algebra of Standard VectorsThe Unit Coordinate VectorsA vector of length 1 is called a unit vectorso that in R2, the unit vectors are vectorswith tips on the unit circle.
73. 73. Algebra of Standard VectorsThe Unit Coordinate VectorsA vector of length 1 is called a unit vectorso that in R2, the unit vectors are vectorswith tips on the unit circle.
74. 74. Algebra of Standard Vectors j = <0, 1>The Unit Coordinate VectorsA vector of length 1 is called a unit vectorso that in R2, the unit vectors are vectors i = <1, 0>with tips on the unit circle.Let i = <1, 0> and j = <0, 1> be the unit vectors alongthe positive coordinate axes as shown.
75. 75. Algebra of Standard Vectors j = <0, 1>The Unit Coordinate VectorsA vector of length 1 is called a unit vectorso that in R2, the unit vectors are vectors i = <1, 0>with tips on the unit circle.Let i = <1, 0> and j = <0, 1> be the unit vectors alongthe positive coordinate axes as shown.Then a vector <a, b> may be written as<a, b> = a<1, 0> + b<0, 1> = ai + bj.
76. 76. Algebra of Standard Vectors j = <0, 1>The Unit Coordinate VectorsA vector of length 1 is called a unit vectorso that in R2, the unit vectors are vectors i = <1, 0>with tips on the unit circle.Let i = <1, 0> and j = <0, 1> be the unit vectors alongthe positive coordinate axes as shown.Then a vector <a, b> may be written as<a, b> = a<1, 0> + b<0, 1> = ai + bj.
77. 77. Algebra of Standard Vectors j = <0, 1>The Unit Coordinate VectorsA vector of length 1 is called a unit vectorso that in R2, the unit vectors are vectors i = <1, 0>with tips on the unit circle.Let i = <1, 0> and j = <0, 1> be the unit vectors alongthe positive coordinate axes as shown.Then a vector <a, b> may be written as<a, b> = a<1, 0> + b<0, 1> = ai + bj.For example <3, 2> = 3i + 2j and that <–4, 1> = –4i +1j.
78. 78. Algebra of Standard Vectors j = <0, 1>The Unit Coordinate VectorsA vector of length 1 is called a unit vectorso that in R2, the unit vectors are vectors i = <1, 0>with tips on the unit circle.Let i = <1, 0> and j = <0, 1> be the unit vectors alongthe positive coordinate axes as shown.Then a vector <a, b> may be written as<a, b> = a<1, 0> + b<0, 1> = ai + bj.For example <3, 2> = 3i + 2j and that <–4, 1> = –4i +1j.We may track vector calculation with i and j.
79. 79. Algebra of Standard Vectors j = <0, 1>The Unit Coordinate VectorsA vector of length 1 is called a unit vectorso that in R2, the unit vectors are vectors i = <1, 0>with tips on the unit circle.Let i = <1, 0> and j = <0, 1> be the unit vectors alongthe positive coordinate axes as shown.Then a vector <a, b> may be written as<a, b> = a<1, 0> + b<0, 1> = ai + bj.For example <3, 2> = 3i + 2j and that <–4, 1> = –4i +1j.We may track vector calculation with i and j.For example <3, –4> – <1, –5> = (3i – 4j) – (i – 5j)
80. 80. Algebra of Standard Vectors j = <0, 1>The Unit Coordinate VectorsA vector of length 1 is called a unit vectorso that in R2, the unit vectors are vectors i = <1, 0>with tips on the unit circle.Let i = <1, 0> and j = <0, 1> be the unit vectors alongthe positive coordinate axes as shown.Then a vector <a, b> may be written as<a, b> = a<1, 0> + b<0, 1> = ai + bj.For example <3, 2> = 3i + 2j and that <–4, 1> = –4i +1j.We may track vector calculation with i and j.For example <3, –4> – <1, –5> = (3i – 4j) – (i – 5j)= 3i – 4j – i + 5j = 2i + j = <2, 1>.
81. 81. Algebra of Standard Vectors j = <0, 1>The Unit Coordinate VectorsA vector of length 1 is called a unit vectorso that in R2, the unit vectors are vectors i = <1, 0>with tips on the unit circle.Let i = <1, 0> and j = <0, 1> be the unit vectors alongthe positive coordinate axes as shown.Then a vector <a, b> may be written as<a, b> = a<1, 0> + b<0, 1> = ai + bj.For example <3, 2> = 3i + 2j and that <–4, 1> = –4i +1j.We may track vector calculation with i and j.For example <3, –4> – <1, –5> = (3i – 4j) – (i – 5j)= 3i – 4j – i + 5j = 2i + j = <2, 1>.In many vector related calculations, it’s easier to keep
82. 82. Algebra of Standard VectorsAll the above terminology andoperations may be applied to 3Dvectors.
83. 83. Algebra of Standard VectorsAll the above terminology andoperations may be applied to 3Dvectors. A 3D vector in thestandard position is a vector whois based at (0, 0, 0).
84. 84. Algebra of Standard VectorsAll the above terminology andoperations may be applied to 3Dvectors. A 3D vector in thestandard position is a vector whois based at (0, 0, 0).We write v = <a, b, c> if the tip ofa 3D standard position vector v inthe is (a, b, c).
85. 85. Algebra of Standard VectorsAll the above terminology andoperations may be applied to 3D z+vectors. A 3D vector in the ystandard position is a vector who 1 2is based at (0, 0, 0). x –3We write v = <a, b, c> if the tip ofa 3D standard position vector v in u = <2,1,–3>the is (a, b, c).
86. 86. Algebra of Standard VectorsAll the above terminology andoperations may be applied to 3D z+vectors. A 3D vector in the v = <–1,–2,1> ystandard position is a vector who 1 2is based at (0, 0, 0). 1 –2 x –3We write v = <a, b, c> if the tip of –1a 3D standard position vector v in u = <2,1,–3>the is (a, b, c).
87. 87. Algebra of Standard VectorsAll the above terminology andoperations may be applied to 3D z+vectors. A 3D vector in the v = <–1,–2,1> ystandard position is a vector who 1 2is based at (0, 0, 0). 1 –2 x –3We write v = <a, b, c> if the tip of –1a 3D standard position vector v in u = <2,1,–3>the is (a, b, c).The zero vector is 0 = <0, 0, 0>.
88. 88. Algebra of Standard VectorsAll the above terminology andoperations may be applied to 3D z +vectors. A 3D vector in the v = <–1,–2,1> ystandard position is a vector who 1 2is based at (0, 0, 0). 1 –2 x –3We write v = <a, b, c> if the tip of –1a 3D standard position vector v in u = <2,1,–3>the is (a, b, c).The zero vector is 0 = <0, 0, 0>.If A = (a, b, c), B = (r, s, t), then the vector AB in thestandard position is B – A = <r – a, s – b, t – c>.
89. 89. Algebra of Standard VectorsAll the above terminology andoperations may be applied to 3D z +vectors. A 3D vector in the v = <–1,–2,1> ystandard position is a vector who 1 2is based at (0, 0, 0). 1 –2 x –3We write v = <a, b, c> if the tip of –1a 3D standard position vector v in u = <2,1,–3>the is (a, b, c).The zero vector is 0 = <0, 0, 0>.If A = (a, b, c), B = (r, s, t), then the vector AB in thestandard position is B – A = <r – a, s – b, t – c>.Scalar-multiplication, vector addition, and the norm(magnitude) are defined similarly as in the case 2Dvectors.
90. 90. Algebra of Standard VectorsScalar MultiplicationLet u = <a, b, c> and λ be a real number, thenλu = λ<a, b, c> = <λa, λb, λc>.This corresponds to stretching u by a factor of λ.
91. 91. Algebra of Standard VectorsScalar MultiplicationLet u = <a, b, c> and λ be a real number, thenλu = λ<a, b, c> = <λa, λb, λc>.This corresponds to stretching u by a factor of λ.Vector Addition (Parallelogram Rule)Let u = <a, b, c>, v = <r, s, t>, thenu + v = <a + r, b + s, c + t>.
92. 92. Algebra of Standard VectorsScalar MultiplicationLet u = <a, b, c> and λ be a real number, thenλu = λ<a, b, c> = <λa, λb, λc>.This corresponds to stretching u by a factor of λ.Vector Addition (Parallelogram Rule)Let u = <a, b, c>, v = <r, s, t>, thenu + v = <a + r, b + s, c + t>.The norm (magnitude or length) of uis |u| = √a2 + b2 + c2 ( = √Δx2 + Δy2 + Δx2).
93. 93. Algebra of Standard VectorsScalar MultiplicationLet u = <a, b, c> and λ be a real number, thenλu = λ<a, b, c> = <λa, λb, λc>.This corresponds to stretching u by a factor of λ.Vector Addition (Parallelogram Rule)Let u = <a, b, c>, v = <r, s, t>, thenu + v = <a + r, b + s, c + t>.The norm (magnitude or length) of uis |u| = √a2 + b2 + c2 ( = √Δx2 + Δy2 + Δx2).We write the coordinate unit vectors asi = <1, 0, 0>, j = <0, 1, 0> and k = <0, 0, 1>,then any vector <a, b, c> may be written as<a, b, c> = a<1, 0, 0> + b<0, 1, 0> + c<0, 0, 1><a, b, c> = ai + bj + ck
94. 94. Algebra of Standard VectorsExample D. Let u = <–1, 1, 2>, v = <1, 3, 3>.Find 3u – 2v, write 3u – 2v in terms of i , j, k, andfind |3u – 2v|.3u – 2v = 3<–1, 1, 2> – 2<1, 3, 3> = <–3, 3, 6> + <–2, –6, –6> = <–5, –3, 0> = –5i – 3j + 0kIn particular |3u – 2v| = √25 + 9 + 0 = √34