Slide 1 - Vectors: Properties of Cross Product

  • The cross product of two vectors is a vector.
  • It is denoted by the symbol “×” or “⨯”.
  • The cross product of two vectors A and B is written as A × B.
  • The magnitude of the cross product is given by |A × B| = |A| |B| sinθ, where θ is the angle between the vectors.
  • The direction of the cross product is perpendicular to both A and B according to the right-hand rule.

Slide 2 - Cross Product of Two Vectors

Given vectors A = (A1, A2, A3) and B = (B1, B2, B3), the cross product A × B is calculated as follows:

  • A × B = (A2B3 - A3B2) i + (A3B1 - A1B3) j + (A1B2 - A2B1) k.
  • The cross product is a vector quantity represented by a bold symbol.

Slide 3 - Properties of Cross Product

The cross product of two vectors has the following properties:

  1. Non-commutativity: A × B = -B × A
  1. Associativity with scalar multiplication: k(A × B) = (kA) × B = A × (kB), where k is any scalar.
  1. Distributivity: A × (B + C) = (A × B) + (A × C)

Slide 4 - Geometrical Interpretation

The cross product has a geometrical interpretation:

  • The magnitude of A × B gives the area of the parallelogram formed by A and B.
  • The direction of A × B is perpendicular to the plane containing A and B.

Slide 5 - Unit Vectors

Unit vectors are vectors with a magnitude of 1.

  • i, j, and k are the unit vectors along the x, y, and z axes, respectively.
  • They are mutually perpendicular and form a right-handed coordinate system.

Slide 6 - Cross Product in Terms of Unit Vectors

The cross product of two vectors A and B can also be expressed using unit vectors:

  • A × B = (A1i + A2j + A3k) × (B1i + B2j + B3k)
  • = (A2B3 - A3B2) i + (A3B1 - A1B3) j + (A1B2 - A2B1) k

Slide 7 - Dot Product and Cross Product

The dot product and cross product are two different operations involving vectors.

  • The dot product yields a scalar, while the cross product results in a vector.
  • The dot product is commutative and distributive, while the cross product is not commutative.
  • Both operations have geometric interpretations and applications.

Slide 8 - Examples: Cross Product Calculation

Example 1: Given vectors A = (2, -3, 4) and B = (1, -2, 0), calculate A × B. Solution: A × B = (A2B3 - A3B2) i + (A3B1 - A1B3) j + (A1B2 - A2B1) = (4 * (-2) - 0 * (-3)) i + (0 * 1 - 2 * 4) j + (2 * (-2) - 1 * (-3)) k = -8i - 8j + 1k

Slide 9 - Examples: Magnitude of Cross Product

Example 2: Given vectors A = (2, -3, 4) and B = (1, -2, 0), find |A × B|. Solution: |A × B| = |A| |B| sinθ = √(2^2 + (-3)^2 + 4^2) * √(1^2 + (-2)^2 + 0^2) * sinθ = √29 * √5 * sinθ

Slide 10 - Examples: Direction of Cross Product

Example 3: Given vectors A = (2, -3, 4) and B = (1, -2, 0), determine the direction of A × B. Solution: The direction of A × B is perpendicular to both A and B according to the right-hand rule. Note: The remaining slides will be covered in subsequent responses.

  1. Vectors - Properties of Cross Product
  • The cross product of two vectors is a vector.
  • It is denoted by the symbol “×” or “⨯”.
  • The cross product of two vectors A and B is written as A × B.
  • The magnitude of the cross product is given by |A × B| = |A| |B| sinθ, where θ is the angle between the vectors.
  • The direction of the cross product is perpendicular to both A and B according to the right-hand rule.
  1. Cross Product of Two Vectors Given vectors A = (A1, A2, A3) and B = (B1, B2, B3), the cross product A × B is calculated as follows:
  • A × B = (A2B3 - A3B2) i + (A3B1 - A1B3) j + (A1B2 - A2B1).
  • The cross product is a vector quantity represented by a bold symbol.
  1. Properties of Cross Product The cross product of two vectors has the following properties:
  • Non-commutativity: A × B = -B × A.
  • Associativity with scalar multiplication: k(A × B) = (kA) × B = A × (kB), where k is any scalar.
  • Distributivity: A × (B + C) = (A × B) + (A × C).
  1. Geometrical Interpretation The cross product has a geometrical interpretation:
  • The magnitude of A × B gives the area of the parallelogram formed by A and B.
  • The direction of A × B is perpendicular to the plane containing A and B.
  1. Unit Vectors Unit vectors are vectors with a magnitude of 1.
  • i, j, and k are the unit vectors along the x, y, and z axes, respectively.
  • They are mutually perpendicular and form a right-handed coordinate system.
  1. Cross Product in Terms of Unit Vectors The cross product of two vectors A and B can also be expressed using unit vectors:
  • A × B = (A1i + A2j + A3k) × (B1i + B2j + B3k)
  • = (A2B3 - A3B2) i + (A3B1 - A1B3) j + (A1B2 - A2B1) k
  1. Dot Product and Cross Product The dot product and cross product are two different operations involving vectors.
  • The dot product yields a scalar, while the cross product results in a vector.
  • The dot product is commutative and distributive, while the cross product is not commutative.
  • Both operations have geometric interpretations and applications.
  1. Examples: Cross Product Calculation Example 1: Given vectors A = (2, -3, 4) and B = (1, -2, 0), calculate A × B. Solution: A × B = (A2B3 - A3B2) i + (A3B1 - A1B3) j + (A1B2 - A2B1) = (4 * (-2) - 0 * (-3)) i + (0 * 1 - 2 * 4) j + (2 * (-2) - 1 * (-3)) k = -8i - 8j + 1k
  1. Examples: Magnitude of Cross Product Example 2: Given vectors A = (2, -3, 4) and B = (1, -2, 0), find |A × B|. Solution: |A × B| = |A| |B| sinθ = √(2^2 + (-3)^2 + 4^2) * √(1^2 + (-2)^2 + 0^2) * sinθ = √29 * √5 * sinθ
  1. Examples: Direction of Cross Product Example 3: Given vectors A = (2, -3, 4) and B = (1, -2, 0), determine the direction of A × B. Solution: The direction of A × B is perpendicular to both A and B according to the right-hand rule.
  1. Examples: Cross Product Application Example 4: A force of 5 N is applied to the right along vector A = (2, 0, 0). Another force of 8 N is applied upward along vector B = (0, 0, 3). Find the resultant force and its direction. Solution: To find the resultant force, calculate the cross product of A and B: Resultant force = A × B = (2i + 0j + 0k) × (0i + 0j + 3k) = (0 * 0 - 0 * 0) i + (0 * 0 - 2 * 3) j + (2 * 0 - 0 * 0) k = -6j The magnitude of the resultant force is |Resultant force| = |-6j| = 6 N. Since the resultant force is in the downward direction (negative y-axis), the direction can be represented as downward or -y direction.
  1. Cross Product in 2D The cross product can also be applied in 2D situations, where vectors lie in the xy plane. For vectors A = (A1, A2, 0) and B = (B1, B2, 0), the cross product A × B can be calculated as follows:
  • A × B = (A2 * 0 - 0 * B2) i + (0 * B1 - A1 * 0) j + (A1 * B2 - A2 * B1) k = 0i + 0j + (A1 * B2 - A2 * B1) k = (A1 * B2 - A2 * B1) k The cross product in 2D results in a vector perpendicular to the xy plane.
  1. Cross Product in Applications The cross product has various applications in different fields, such as:
  • Mechanics: In rotational motion, torque can be calculated using the cross product of the force and the position vector.
  • Electromagnetism: The magnetic field created by a current-carrying wire can be determined using the cross product of the current and the position vector.
  • Physics: The angular momentum of a rotating object can be calculated using the cross product of the position vector and the linear momentum vector.
  1. Cross Product and Determinants The cross product of two vectors A and B can also be expressed using determinants:
  • A × B = | i j k | | A1 A2 A3 | | B1 B2 B3 | The determinant provides a convenient way to calculate the cross product without calculating each component separately.
  1. Cross Product and Scalar Triple Product The cross product of three vectors A, B, and C can be related to the scalar triple product:
  • A × (B × C) = (A · C)B - (A · B)C The scalar triple product helps simplify the calculation and manipulation of cross products in certain situations.
  1. Examples: Cross Product Calculation Using Determinants Example 5: Given vectors A = (1, 2, -1) and B = (3, -1, 2), calculate A × B using determinants. Solution: A × B = | i j k | | 1 2 -1 | | 3 -1 2 | = i(2 * 2 - (-1) * (-1)) - j(1 * 2 - (-1) * 3) + k(1 * (-1) - 2 * 3) = 5i - 5j + 7k
  1. Examples: Dot Product and Cross Product Example 6: Given vectors A = (2, -3, 4) and B = (1, -2, 0), find the dot product A · (A × B). Solution: The dot product can be found using the cross product: A · (A × B) = (2i - 3j + 4k) · (-8i - 8j + k) = 2 * (-8) + (-3) * (-8) + 4 * 1 = -16 + 24 + 4 = 12
  1. Examples: Angle Between Vectors Example 7: Given vectors A = (2, -3, 4) and B = (1, -2, 0), find the angle between A and B. Solution: The angle can be found using the cross product and dot product: |A × B| = |A| |B| sinθ |A × B| = √29 * √5 * sinθ |A| = √(2^2 + (-3)^2 + 4^2) = √29 |B| = √(1^2 + (-2)^2 + 0^2) = √5 Using the dot product: A · B = |A| |B| cosθ |A| |B| cosθ = √29 * √5 * cosθ Substituting the values:

12 = √29 * √5 * cosθ Solving for cosθ: cosθ = 12 / (√29 * √5) Taking the inverse cosine: θ = cos^(-1)(12 / (√29 * √5))

  1. Summary Key points to remember about cross product:
  • The cross product of two vectors is a vector quantity.
  • It is calculated using the formula A × B = (A2B3 - A3B2) i + (A3B1 - A1B3) j + (A1B2 - A2B1).
  • The cross product has properties like non-commutativity and associativity with scalar multiplication.
  • Its magnitude gives the area of the parallelogram formed by the vectors, and its direction is perpendicular to the plane containing the vectors.
  • The cross product can be expressed using unit vectors, determinants, and scalar triple products.
  • Applications include mechanics, electromagnetism, and physics.
  1. Questions
  1. Calculate the cross product of A = (3, -1, 2) and B = (-2, 0, 1).
  1. Find the angle between vectors A = (2, -4, 1) and B = (1, 3, 2).
  1. Calculate the dot product of A = (4, -3, 2) and B = (2, 5, -1).
  1. In a 2D plane, A = (3, 4, 0) and B = (2, -1, 0). Find A × B.
  1. Determine the direction of the cross product of A = (1, 0, -3) and B = (0, 4, 2). Note: Please attempt these questions on your own and discuss the solutions in the next lecture.