Lesson 2: Vectors and the Dot Product

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A vector has magnitude and direction. There is an algebra and geometry of vectors which makes addition, subtraction, and scaling well-defined.

The scalar or dot product of vectors measures the angle between them, in a way. It's useful to show if two vectors are perpendicular or parallel.

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Lesson 2: Vectors and the Dot Product

  1. 1. Section 9.2–3 Vectors and the Dot Product Math 21a February 6, 2008 Announcements The MQC is open: Sun-Thu 8:30pm–10:30pm, SC 222 Homework for Friday 2/8: Section 9.2: 4, 6, 26, 33, 34 Section 9.3: 10, 18, 24, 25, 34 Section 9.4: 1*
  2. 2. Outline Vectors Algebra of Vectors Components Standard basis vectors Length The Dot Product Work Concept Properties Uses
  3. 3. What is a vector? Definition A vector is something that has magnitude and direction We denote vectors by boldface (v) or little arrows (v ). One is good for print, one for script Given two points A and B in flatland or spaceland, the vector which starts at A and ends at B is called the displacement −→ vector AB. Two vectors are equal if they have the same magnitude and direction (they need not overlap) B D v u A C
  4. 4. Vector or scalar? Definition A scalar is another name for a real number. Example Which of these are vectors or scalars? (i) Cost of a theater ticket (ii) The current in a river (iii) The initial flight path from Boston to New York (iv) The population of the world
  5. 5. Vector or scalar? Definition A scalar is another name for a real number. Example Which of these are vectors or scalars? (i) Cost of a theater ticket scalar (ii) The current in a river vector (iii) The initial flight path from Boston to New York vector (iv) The population of the world scalar
  6. 6. Vector addition Definition If u and v are vectors positioned so the initial point of v is the terminal point of u, the sum u + v is the vector whose initial point is the initial point of u and whose terminal point is the terminal point of v. u v v u+v u+v v u u The triangle law The parallelogram law
  7. 7. Opposite and difference Definition Given vectors u and v, the opposite of v is the vector −v that has the same length as v but points in the opposite direction the difference u − v is the sum u + (−v) v u −v u−v
  8. 8. Scaling vectors Definition If c is a nonzero scalar and v is a vector, the scalar multiple cv is the vector whose length is |c| times the length of v direction is the same as v if c > 0 and opposite v if c < 0 If c = 0, cv = 0. v
  9. 9. Scaling vectors Definition If c is a nonzero scalar and v is a vector, the scalar multiple cv is the vector whose length is |c| times the length of v direction is the same as v if c > 0 and opposite v if c < 0 If c = 0, cv = 0. 2v v
  10. 10. Scaling vectors Definition If c is a nonzero scalar and v is a vector, the scalar multiple cv is the vector whose length is |c| times the length of v direction is the same as v if c > 0 and opposite v if c < 0 If c = 0, cv = 0. v 1 −2v
  11. 11. Properties Theorem Given vectors a, b, and c and scalars c and d, we have 1. a + b = b + a 5. c(a + b) = ca + cb 2. a + (b + c) = (a + b) + c 6. (c + d)a = ca + da 3. a + 0 = a 7. (cd)a = c(da) 4. a + (−a) = 0 8. 1a = a These can be verified geometrically.
  12. 12. Components defined Definition Given a vector a, it’s often useful to move the tail to O and measure the coordinates of the head. These are called the components of a, and we write them like this: a = a1 , a2 , a3 or just two components if the vector is the plane. Note the angle brackets!
  13. 13. Components defined Definition Given a vector a, it’s often useful to move the tail to O and measure the coordinates of the head. These are called the components of a, and we write them like this: a = a1 , a2 , a3 or just two components if the vector is the plane. Note the angle brackets! Given a point P in the plane or space, the position vector of −→ P is the vector OP.
  14. 14. Components defined Definition Given a vector a, it’s often useful to move the tail to O and measure the coordinates of the head. These are called the components of a, and we write them like this: a = a1 , a2 , a3 or just two components if the vector is the plane. Note the angle brackets! Given a point P in the plane or space, the position vector of −→ P is the vector OP. Fact −→ Given points A(x1 , y1 , z1 ) and B(x2 , y2 , z2 ) in space, the vector AB has components −→ AB = x2 − x1 , y2 − y1 , z2 − z1
  15. 15. Vector algebra in components Theorem If a = a1 , a2 , a3 and b = b1 , b2 , b3 , and c is a scalar, then a + b = a1 + b1 , a2 + b2 , a3 + b3 a − b = a1 − b1 , a2 − b2 , a3 − b3 ca = ca1 , ca2 , ca3
  16. 16. Useful vectors Definition We define the standard basis vectors i = 1, 0, 0 , j = 0, 1, 0 , ı ˆ ˆ k = 0, 0, 1 . In script, they’re often written as ˆ, , k.
  17. 17. Useful vectors Definition We define the standard basis vectors i = 1, 0, 0 , j = 0, 1, 0 , ı ˆ ˆ k = 0, 0, 1 . In script, they’re often written as ˆ, , k. Fact Any vector a can be written as a linear combination of the standard basis vectors a1 , a2 , a3 = a1 i + a2 j + a3 k.
  18. 18. Length Definition Given a vector v, its length is the distance between its initial and terminal points.
  19. 19. Length Definition Given a vector v, its length is the distance between its initial and terminal points. Fact The length of a vector is the square root of the sum of the squares of its components: | a1 , a2 , a3 | = 2 2 2 a1 + a2 + a3
  20. 20. Early vector users Caspar Wessel (Norwegian and Danish, 1745–1818) Jean Robert Argand (French 1768–1822), Carl Friedrich Gauss (German, 1777–1855) Sir William Rowan Hamilton (Irish, 1805–1865)
  21. 21. Outline Vectors Algebra of Vectors Components Standard basis vectors Length The Dot Product Work Concept Properties Uses
  22. 22. Definition Work is the energy needed to move an object by a force.
  23. 23. Definition Work is the energy needed to move an object by a force. If the force is expressed as a vector F and the displacement a vector D, the work is W = |F| |D| cos θ where θ is the angle between the vectors. θ D Work is |F| times this distance F
  24. 24. Definition If a and b are any two vectors in the plane or in space, the dot product (or scalar product) between them is the quantity a · b = |a| |b| cos θ, where θ is the angle between them.
  25. 25. Definition If a and b are any two vectors in the plane or in space, the dot product (or scalar product) between them is the quantity a · b = |a| |b| cos θ, where θ is the angle between them. Another way to say this is that a · b is |b| times the length of the projection of a onto b. a a · b is |b| times this length b
  26. 26. Geometric properties of the dot product Fact Two vectors are perpendicular or orthogonal if their dot π product is zero (i.e., cos θ = 90◦ = ) 2
  27. 27. Geometric properties of the dot product Fact Two vectors are perpendicular or orthogonal if their dot π product is zero (i.e., cos θ = 90◦ = ) 2 The law of cosines can be expressed as |a + b|2 = |a|2 + |b|2 − 2 |a| |b| cos θ = |a|2 + |b|2 − 2a · b
  28. 28. Geometric properties of the dot product Fact Two vectors are perpendicular or orthogonal if their dot π product is zero (i.e., cos θ = 90◦ = ) 2 The law of cosines can be expressed as |a + b|2 = |a|2 + |b|2 − 2 |a| |b| cos θ = |a|2 + |b|2 − 2a · b In components, if a = a1 , a2 , a3 and b = b1 , b2 , b3 , then a · b = a1 b1 + a2 b2 + a3 b3
  29. 29. More geometric properties of the dot product Fact The angle between two nonzero vectors a and b is given by a·b cos θ = , |a| |b| where θ is taken to be between 0 and π. Fact The angle between two nonzero vectors a and b is acute if a · b > 0
  30. 30. More geometric properties of the dot product Fact The angle between two nonzero vectors a and b is given by a·b cos θ = , |a| |b| where θ is taken to be between 0 and π. Fact The angle between two nonzero vectors a and b is acute if a · b > 0 obtuse if a · b < 0
  31. 31. More geometric properties of the dot product Fact The angle between two nonzero vectors a and b is given by a·b cos θ = , |a| |b| where θ is taken to be between 0 and π. Fact The angle between two nonzero vectors a and b is acute if a · b > 0 obtuse if a · b < 0 right if a · b = 0;
  32. 32. More geometric properties of the dot product Fact The angle between two nonzero vectors a and b is given by a·b cos θ = , |a| |b| where θ is taken to be between 0 and π. Fact The angle between two nonzero vectors a and b is acute if a · b > 0 obtuse if a · b < 0 right if a · b = 0;
  33. 33. More geometric properties of the dot product Fact The angle between two nonzero vectors a and b is given by a·b cos θ = , |a| |b| where θ is taken to be between 0 and π. Fact The angle between two nonzero vectors a and b is acute if a · b > 0 obtuse if a · b < 0 right if a · b = 0; The vectors are parallel if a · b = ± |a| |b|. b is a positive multiple of a if a · b = |a| |b|
  34. 34. More geometric properties of the dot product Fact The angle between two nonzero vectors a and b is given by a·b cos θ = , |a| |b| where θ is taken to be between 0 and π. Fact The angle between two nonzero vectors a and b is acute if a · b > 0 obtuse if a · b < 0 right if a · b = 0; The vectors are parallel if a · b = ± |a| |b|. b is a positive multiple of a if a · b = |a| |b| b is a negative multiple of a if a · b = − |a| |b|
  35. 35. Examples Example Find the sum of the following pairs of vectors geometrically and algebraically. (i) a = 3, −1 and b = −2, 4 (ii) a = 0, 1, 2 and b = 0, 0, −3 What is the angle between the two vectors in each case?
  36. 36. Examples Example Find the sum of the following pairs of vectors geometrically and algebraically. (i) a = 3, −1 and b = −2, 4 (ii) a = 0, 1, 2 and b = 0, 0, −3 What is the angle between the two vectors in each case? Solution √ √ (i) a + b = 1, 3 , |a| = 10, |b| = 20. So a·b −6 − 4 −10 1 3π cos θ = = √ √ = √ √ = − √ =⇒ θ = |a| |b| 10 20 10 20 2 4 (ii) a + b = 0, 1, −1 , while 0+0−6 2 cos θ = √ √ = −√ 5 9 5
  37. 37. Properties Fact If a, b and c are vectors are c is a scalar, then 1. a · a = |a|2 4. (ca) · b = c(a · b) = a · (cb) 2. a · b = b · a 3. a · (b + c) = a · b + a · c 5. 0 · a = 0
  38. 38. Example The dot product can be used to measure how similar two vectors are. Consider it a compatibility index. If two vectors point in approximately the same direction, we get a positive dot product. If two vectors are orthogonal, we get a zero dot product. If two vectors point in approximately opposite directions, we get a negative dot product. Consider the following categories, 1. Football 2. Sushi 3. Classical music Now create a vector in R3 rating your preference in each category from −5 to 5, where −5 expresses extreme dislike and 5 expresses adoration. Dot your vector with your neighbor’s.
  39. 39. Example Fifi, a poodle, drags her owner along a sidewalk that is 200 meters long. If Fifi exerts a force of two newtons on the leash, and the leash is at an angle 45◦ from the ground, how much work does Fifi do?

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