Vector Operations and Properties
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Questions and Answers

What is the result of subtracting vector b from vector a?

  • a - b = (a1 - b1, a2 - b2) (correct)
  • a - b = (a1 * b1, a2 * b2)
  • a - b = (a1 + b1, a2 + b2)
  • a - b = (-a1 + b1, -a2 + b2)

    • What is the commutative property of scalar multiplication?

  • k(a + b) = ka + kb
  • kl(a) = k(la)
  • (a + b)k = ak + bk
  • k(a) = ak (correct)

    • What is the geometric interpretation of the dot product?

  • a · b = |a| + |b|
  • a · b = |a||b|sin(θ)
  • a · b = |a||b|cos(θ) (correct)
  • a · b = |a| - |b|

    • What is the result of adding vector a and vector b?

    <p>a + b = (a1 + b1, a2 + b2)</p> Signup and view all the answers

    What is the distributive property of scalar multiplication?

    <p>k(a + b) = ka + kb</p> Signup and view all the answers

    What is the property of the dot product that states a · b = b · a?

    <p>Commutative property</p> Signup and view all the answers

    What is the result of multiplying a vector by a scalar?

    <p>A new vector with the same direction</p> Signup and view all the answers

    What is the purpose of the dot product?

    <p>To measure the similarity between two vectors</p> Signup and view all the answers

    What is the associative property of scalar multiplication?

    <p>(k * l) * a = k * (l * a)</p> Signup and view all the answers

    What is the result of subtracting vector b from vector a?

    <p>a - b</p> Signup and view all the answers

    What is the positive definiteness property of the dot product?

    <p>a · a ≥ 0 and a · a = 0 if a = 0</p> Signup and view all the answers

    What is the distributive property of the dot product?

    <p>a · (b + c) = a · b + a · c</p> Signup and view all the answers

    Study Notes

    Vector Operations

    • Addition: Two or more vectors can be added by adding corresponding components.
      • Example: a = <a1, a2> and b = <b1, b2> then a + b = <a1 + b1, a2 + b2>
    • Subtraction: One vector can be subtracted from another by subtracting corresponding components.
      • Example: a = <a1, a2> and b = <b1, b2> then a - b = <a1 - b1, a2 - b2>
    • Equality: Two vectors are equal if and only if their corresponding components are equal.
      • Example: a = <a1, a2> and b = <b1, b2> then a = b if and only if a1 = b1 and a2 = b2

    Scalar Multiplication

    • Scalar multiplication: A vector can be multiplied by a scalar (number) to change its magnitude and/or direction.
      • Example: a = <a1, a2> and k is a scalar, then ka = <ka1, ka2>
    • Properties:
      • Distributive property: k(a + b) = ka + kb
      • Associative property: (kl)a = k(la)
      • Commutative property: k(a + b) = (ka) + (kb)

    Dot Product

    • Dot product (Scalar Product): The dot product of two vectors is a scalar value that can be used to find the angle between two vectors.
      • Example: a = <a1, a2> and b = <b1, b2> then a · b = a1b1 + a2b2
    • Properties:
      • Commutative property: a · b = b · a
      • Distributive property: a · (b + c) = a · b + a · c
      • Geometric interpretation: The dot product of two vectors can be used to find the angle between them: a · b = |a||b|cos(θ)

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    Description

    Test your understanding of vector operations, including addition, subtraction, equality, scalar multiplication, and dot product. Learn about the properties and geometric interpretations of these operations.

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