Vectors - Problems - Expanding vector expressions with dot and cross

  • Introduction to vector expressions
  • Product of vectors
  • Dot product in vector expressions
  • Cross product in vector expressions
  • Expanding vector expressions
  • Example 1: Expanding a dot product expression
  • Example 2: Expanding a cross product expression
  • Example 3: Expanding a combination of dot and cross products
  • Summary
  • Practice questions

Introduction to vector expressions

  • Vectors are quantities that have both magnitude and direction.
  • Vector expressions involve the use of vectors and algebraic operations.
  • Vector expressions can be expanded to simplify complex equations.

Product of vectors

  • The product of two vectors can be determined using two different operations:
    • Dot product (scalar product)
    • Cross product (vector product)
  • Dot product yields a scalar value, while cross product yields a vector value.

Dot product in vector expressions

  • The dot product of two vectors, A and B, can be denoted as A · B.
  • Dot product is calculated as the product of magnitudes and the cosine of the angle between the vectors.
  • Dot product can simplify vector expressions by eliminating cross terms.

Cross product in vector expressions

  • The cross product of two vectors, A and B, can be denoted as A x B.
  • Cross product is calculated using the determinant of a 3x3 matrix.
  • Cross product can be used to find orthogonal vectors and calculate areas.

Expanding vector expressions

  • Expanding vector expressions involves distributing the operations to each vector/component.
  • Distribute dot product by multiplying each corresponding component and summing.
  • Distribute cross product by calculating the determinant of the resulting matrix.

Example 1: Expanding a dot product expression

  • Given: A = (2, 5) and B = (3, -1)
  • Expand the expression A · B:
    • A · B = (2 * 3) + (5 * -1)
    • A · B = 6 - 5
    • A · B = 1

Example 2: Expanding a cross product expression

  • Given: A = (3, -2, 4) and B = (1, 5, -2)
  • Expand the expression A x B:
    • A x B = ((-2 * -2) - (4 * 5))i - ((3 * -2) - (4 * 1))j + ((3 * 5) - (-2 * 1))k
    • A x B = (4 - 20)i - (-6 - 4)j + (15 + 2)k
    • A x B = -16i + 10j + 17k

Example 3: Expanding a combination of dot and cross products

  • Given: A = (1, 2) and B = (3, 4) and C = (5, 6)
  • Expand the expression A · (B x C):
    • A · (B x C) = A · [(3, 4) x (5, 6)]
    • A · (B x C) = A · (-8, 2)
    • A · (B x C) = (1 * -8) + (2 * 2)
    • A · (B x C) = -8 + 4
    • A · (B x C) = -4

Summary

  • Vector expressions involve the use of vectors and algebraic operations.
  • Dot product and cross product are two important operations in vector expressions.
  • Expanding vector expressions simplifies complex equations.
  • Dot product yields a scalar value, while cross product yields a vector value.

Practice questions

  1. Expand the expression (2, 3) · (4, -1).
  1. Expand the expression (6i + 2j) x (3i - 5j).
  1. Expand the expression (3i + 4j) · [(2i + 3j) x (5i - 2j)].
  1. Dot product in vector expressions:
  • The dot product of two vectors, A and B, is given by the formula: A · B = |A| |B| cos(θ), where θ is the angle between the two vectors.
  • The dot product can also be calculated by multiplying the corresponding components of the vectors and summing them.
  • The dot product has several important properties, including commutativity and distributivity over vector addition.
  1. Properties of dot product:
  • Commutativity: A · B = B · A
  • Distributivity over vector addition: A · (B + C) = A · B + A · C
  • Dot product with zero vector: A · 0 = 0
  1. Cross product in vector expressions:
  • The cross product of two vectors, A and B, is denoted by A x B.
  • The cross product is a vector that is perpendicular to both A and B.
  • The magnitude of the cross product is given by |A x B| = |A| |B| sin(θ), where θ is the angle between the two vectors.
  • The direction of the cross product is determined by the right-hand rule.
  1. Properties of cross product:
  • Anti-commutativity: A x B = -(B x A)
  • Distributivity over vector addition: A x (B + C) = A x B + A x C
  • Cross product with zero vector: A x 0 = 0
  1. Expanding a dot product expression:
  • Example: A = (2, -1, 4) and B = (3, 2, -1)
    • A · B = (2 * 3) + (-1 * 2) + (4 * -1)
    • A · B = 6 - 2 - 4
    • A · B = 0
  1. Expanding a cross product expression:
  • Example: A = (1, 2, 3) and B = (4, 5, 6)
    • A x B = ((2 * 6) - (3 * 5))i - ((1 * 6) - (3 * 4))j + ((1 * 5) - (2 * 4))k
    • A x B = (12 - 15)i - (6 - 12)j + (5 - 8)k
    • A x B = -3i + 6j - 3k
  1. Expanding a combination of dot and cross products:
  • Example: A = (1, -3) and B = (2, -1) and C = (3, 4)
    • A · (B x C) = A · [(2, -1) x (3, 4)]
    • A · (B x C) = A · (-7, -14)
    • A · (B x C) = (1 * -7) + (-3 * -14)
    • A · (B x C) = -7 + 42
    • A · (B x C) = 35
  1. Summary:
  • Vector expressions can be expanded using dot and cross product operations.
  • Dot product yields a scalar value, while cross product yields a vector value.
  • Dot product is calculated by multiplying corresponding components and summing.
  • Cross product is calculated using the determinant of a 3x3 matrix.
  1. Practice questions:
  1. Expand the expression (2, -1) · (3, 4).
  1. Expand the expression (1i - 2j) x (-3i + 4j).
  1. Expand the expression (2i + 3j) · [(4i - 5j) x (-6i + 7j)].
  1. Conclusion:
  • Understanding vector expressions and expanding them is crucial for solving problems in physics, engineering, and mathematics.
  • Practice using dot and cross products to simplify vector expressions and solve related problems.
  • Remember the properties of dot and cross product for easy manipulation of equations.
  • Keep practicing with more examples to strengthen your understanding of vector expressions.

Expanding vector expressions - Examples

  • Example 1: Expanding a dot product expression
    • Given: A = (2, 3) and B = (4, -1)
    • Expand the expression A · B:
      • A · B = (2 * 4) + (3 * -1)
      • A · B = 8 - 3
      • A · B = 5
  • Example 2: Expanding a cross product expression
    • Given: A = (1, 2, 3) and B = (4, 5, 6)
    • Expand the expression A x B:
      • A x B = ((2 * 6) - (3 * 5))i - ((1 * 6) - (3 * 4))j + ((1 * 5) - (2 * 4))k
      • A x B = (12 - 15)i - (6 - 12)j + (5 - 8)k
      • A x B = -3i + 6j - 3k
  • Example 3: Expanding a combination of dot and cross products
    • Given: A = (2, -1, 4), B = (3, 2, -1), and C = (1, 3, 2)
    • Expand the expression (A x B) · C:
      • (A x B) · C = (-3i + 6j - 3k) · (1i + 3j + 2k)
      • (A x B) · C = (-3 * 1) + (6 * 3) + (-3 * 2)
      • (A x B) · C = -3 + 18 - 6
      • (A x B) · C = 9
  • Example 4: Expanding a dot product expression with variables
    • Given: A = (x, y) and B = (2x, -3y)
    • Expand the expression A · B:
      • A · B = (x * 2x) + (y * -3y)
      • A · B = 2x^2 - 3y^2
  • Example 5: Expanding a cross product expression with variables
    • Given: A = (x, y, z) and B = (3x, -2y, 4z)
    • Expand the expression A x B:
      • A x B = ((y * 4z) - (z * -2y))i - ((x * 4z) - (z * 3x))j + ((x * -2y) - (y * 3x))k
      • A x B = (4yz + 2yz)i - (4xz - 3xz)j + (-2xy - 3xy)k
      • A x B = (6yz)i - (xz)j - (5xy)k
  • Practice question 1: Expand the expression (2, 4) · (1, -3).
  • Practice question 2: Expand the expression (3i - j) x (2i + 5j).
  • Practice question 3: Expand the expression (4i - 2j) · [(3i - j) x (2i + 5j)].
  • Summary:
    • Expanding vector expressions involves distributing the operations to each vector/component.
    • Dot product can be expanded by multiplying corresponding components and summing.
    • Cross product can be expanded using the determinant of a 3x3 matrix.
    • Expanding vector expressions helps simplify complex equations.
    • Practice expanding vector expressions to strengthen understanding and problem-solving skills.