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Introduction to Vectors

Introduction to Vectors. UNIT 1. What is a Vector? . A vector is a directed line segment, can be represented as AB or AB, where A and B are the two endpoints of the line segment. Directed means that the vector has a direction. QUESTION: Which direction is implied for vector AB?.

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Introduction to Vectors

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  1. Introduction to Vectors UNIT 1

  2. What is a Vector? • A vector is a directed line segment, can be represented as AB or AB, where A and B are the two endpoints of the line segment. • Directed means that the vector has a direction. • QUESTION: Which direction is implied for vector AB?

  3. Vectors: Example B A

  4. Vector Quantity • There are exactly two properties that completely characterize a vector: • Direction – which way does the vector point? • Magnitude – the length of the vector, written as |XY| for vector XY Together the direction and magnitude define the vector quantity. • QUESTION: What are some examples of vectors that we are already familiar with?

  5. Equal Vectors • Two vectors are equal vectors if they have both the same direction and magnitude

  6. Three cars on the road are driving in the same direction at the same speed: Do they have equal velocity vectors?

  7. Opposite Vectors: An opposite vector is a vector with the same magnitude as the original, but opposite direction: B In this illustration, AB and BA are opposite vectors. A

  8. Vector Sum • A vector sumA + B is defined as a vector that results from placing the initial point of vector B at the terminal point of vector A: the vector with the same initial point as A and the same terminal point as B is the vector sum. B A A + B

  9. Parallelogram Rule • Another method, called the parallelogram rule, to find A + B is to place vectors A and B so that their initial points coincide, then complete a parallelogram that has A and B as two adjacent sides. The diagonal of the parallelogram with the same initial points is the vector sum:

  10. Parallelogram Rule Example A A + B B

  11. The two methods side by side: B A A + B A A + B B The two methods give identical results for the vector sum

  12. QUESTION • Two forces of 15 newtons and 22 newtons act at a point in the plane. If the angle between the forces is 100°, find the magnitude of the resultant force: 22 100° 15

  13. QUESTION • Two forces of 15 newtons and 22 newtons act at a point in the plane. If the angle between the forces is 100°, find the magnitude of the resultant force: Z 22 22 100° =80 15

  14. Scalar Product • A real number k and a vector U create the vector kU which has the magnitude |k| times the magnitude of U . kU has the same direction as U if k>0, and the opposite direction if k < 0: • So vector 2U would be twice the length of U and would point in the same direction as U does:

  15. Scalar Product Examples U 2U 3U -U -2U -3U

  16. Position Vectors • A position vector is a vector with its initial point at the origin and with its endpoint at (a, b). It is written <a, b>, so U = <a, b> below: (a, b) Y axis U (0, 0) X axis

  17. Direction Angle of <a, b> • The direction angle is the positive angle between the x-axis and a position vector: (a, b) Y axis U Direction Angle (0, 0) X axis

  18. Direction Angle of <a, b> (cont) The direction angle θ satisfies tan θ = b/a, where a ≠ 0: (a, b) Y axis U b (0, 0) X axis a

  19. Magnitude of vector <a, b> • The magnitude of vector <a, b> is given by |U| = (a, b) Y axis U b (0, 0) X axis a

  20. QUESTION • What is the direction angle and magnitude of vector U=<3, -2>? Y axis (0, 0) X axis U (3, -2)

  21. QUESTION • What is the direction angle and magnitude of vector <3, -2>? • ANSWER: Direction angle = 326.3°, magnitude =

  22. Horizontal and Vertical Components of a Vector • The horizontal and vertical components of a vector U are given by: a = |U|cosθ b = |U|sinθ (a, b) Y axis U b = |U|sinθ (0, 0) X axis a = |U|cosθ

  23. QUESTION • Calculate the vertical and horizontal components of a vector with direction angle of 40° and a magnitude of 25. (a, b) Y axis |U|=25 = 40 (0, 0) X axis

  24. QUESTION • Calculate the horizontal and vertical components of a vector with direction angle of 40° and a magnitude of 25. • ANSWER: x = 19.2, y = 16.1 or <19.2, 16.1>

  25. Vector Operations Overview • <a, b> + <c, d> = <a + c, b + d> • k*<a, b> = <ka, kb> • If A = <a1, a2>, then –A = <-a1, -a2> • <a, b> - <c, d> = <a, b> + -<c, d> OR <a, b> + <-c, -d> = <a – c, b – d>

  26. Vector Operations • For any real numbers a, b, c, and d: <a, b> + <c, d> = <a + c, b + d> (a, b) Y axis (a+c, b+d) (0, 0) X axis (c, d)

  27. Vector Operations • For any real numbers a, b, c, and d: <a, b> + <c, d> = <a + c, b + d> <3, 4> + <4, -2> = <7, 2> (3, 4) Y axis (7, 2) (0, 0) X axis (4, -2)

  28. Vector Operations (cont) B If U = <a1, a2> Then –U = <-a1, -a2>

  29. Vector Operations • Scalar multiplication: k*<a,b> = <ka, kb> • Examples: -3*<4, 7> = <-12, -21> 6*<1, 2> = <6, 12> 0*<3, 5> = <0, 0>

  30. Vector Operations (3, 4) U (6, 8) (0, 0) 2U (0, 0)

  31. QUESTION • Consider the vectors shown in the following figure, and perform the operations: U + V -2U 4U – 3V (x, y) (4, 3) Y axis U + V (-2, 1) V U (0, 0) X axis

  32. Vector Subtraction • Vector subtraction is the inverse operation of vector addition and is defined as adding the negative vector: So we have B – A = B + (-A) for all vectors A, B Therefore (see below) B – A = C B A C -A -B

  33. Vector Subtraction • <a, b> - <c, d> = <a, b> + -<c, d> OR <a, b> + <-c, -d> = <a – c, b – d> • Examples: <3, 6> - <4, 4> = <-1, 2> <6, -4> - <-2, -5> = <8, 1> <0, 1> - <4, 10> = <-4, -9>

  34. Vector Subtraction QUESTIONS • QUESTION: Express A as a difference of two vectors. • QUESTION: Express B as a difference of two vectors. B A C

  35. Vector Notation Conventions • Unit Vectors: i = <1, 0>, j=<0, 1> • i, j Form for Vectors: If v = <a, b>, then v = ai + bj

  36. Unit Vectors j • i U = 5i + 3j (5, 3) Y axis j U j j (0, 0) X axis • i • i • i • i • i

  37. QUESTION • Write the vector in the form ai + bj: <-5,8>

  38. QUESTION • Write the vector in the form ai + bj: <-5,8> Answer: If v=<a,b>, then: V=ai + bj, so V = -5i + 8j

  39. Dot Product • The dot product of the two vectors U = <a, b> and V = <c, d> is denoted by U•V, read “U dot V,” is given by U•V = ac + bd. • Examples: <2, 3>•<4, -1> = <6, 4>•<-2, 3> =

  40. Geometric Interpretation of the Dot Product • If θ is the angle between two nonzero vectors U and V, where 0° <θ< 180°, then U•V = |U|*|V|cosθ • Example: <2, 0>•<3, 3> = 6 using <a,b>•<c,d>=ac+bd But it is also true using U•V = |U|*|V|cosθ

  41. Geometric Interpretation of the Dot Product U • V = |U||V|cosθ = (3 (3, 3) Y axis U (2, 0) θ=45 (0, 0) X axis V

  42. Properties of the Dot Product U•V = V•U U•(V+W)=U•V + U•W (U + V)•W = U•W + V•W (kU)•V = k(U•V) = U•(kV) 0•U = 0 U•U = |U|2

  43. The Dot Product Can Be Positive, Zero, or Negative θ < 90: Positive dot product • θ • θ = 90: Zero dot product • θ • θ θ > 90: Negative dot product

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