Lesson 2: Vectors and the Dot Product
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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....

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 Lesson 2: Vectors and the Dot Product Presentation Transcript

  • 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*
  • Outline Vectors Algebra of Vectors Components Standard basis vectors Length The Dot Product Work Concept Properties Uses
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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.
  • 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!
  • 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.
  • 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
  • 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
  • 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.
  • 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.
  • Length Definition Given a vector v, its length is the distance between its initial and terminal points.
  • 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
  • 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)
  • Outline Vectors Algebra of Vectors Components Standard basis vectors Length The Dot Product Work Concept Properties Uses
  • Definition Work is the energy needed to move an object by a force.
  • 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
  • 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.
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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;
  • 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;
  • 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|
  • 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|
  • 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?
  • 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
  • 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
  • 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.
  • 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?