Vector Operations: Vector Addition

Vector Operations

Vector Addition

• Definition: The sum of two or more vectors is a vector that represents the combined effect of the individual vectors.
• Notation: The sum of vectors A and B is denoted as A + B.
• Properties:
  • Commutative: A + B = B + A
  • Associative: (A + B) + C = A + (B + C)
  • Distributive: A + (B + C) = (A + B) + C
• Graphical Representation: The sum of two vectors is the diagonal of the parallelogram formed by the two vectors.

Vector Multiplication

• Scalar Multiplication: Multiplying a vector by a scalar (number) changes its magnitude but not its direction.
• Properties:
  • k(A + B) = kA + kB
  • (k + l)A = kA + lA
• Vector Multiplication Types:
  • Dot Product (Scalar Product): Produces a scalar value.
  • Cross Product (Vector Product): Produces a vector value.

Cross Product (Vector Product)

• Definition: The cross product of two vectors A and B is a vector C that is perpendicular to both A and B.
• Notation: A × B or A ∧ B
• Properties:
  • A × B = - B × A (Anti-commutative)
  • A × (B + C) = A × B + A × C (Distributive)
• Right-Hand Rule: To determine the direction of the cross product, use the right-hand rule: Point your thumb in the direction of A, your index finger in the direction of B, and your middle finger will point in the direction of A × B.

Dot Product (Scalar Product)

• Definition: The dot product of two vectors A and B is a scalar value that represents the amount of "similarity" between the two vectors.
• Notation: A · B or A ⋅ B
• Properties:
  • A · B = B · A (Commutative)
  • A · (B + C) = A · B + A · C (Distributive)
• Geometric Interpretation: The dot product is related to the cosine of the angle between the two vectors.

Vector Projections

• Definition: The projection of a vector A onto a vector B is a vector that represents the component of A in the direction of B.
• Notation: projB(A)
• Formula: projB(A) = (A · B / ||B||²) B
• Properties:
  • projB(A) is parallel to B
  • ||projB(A)|| ≤ ||A||
• Applications: Finding the component of a force in a specific direction, resolving a force into its components.

Vector Operations

Vector Addition

• Vector addition is a vector that represents the combined effect of individual vectors.
• Notation: A + B denotes the sum of vectors A and B.
• Properties:
  • Commutative: A + B = B + A
  • Associative: (A + B) + C = A + (B + C)
  • Distributive: A + (B + C) = (A + B) + C
• Graphical representation: The sum of two vectors is the diagonal of the parallelogram formed by the two vectors.

Vector Multiplication

• Scalar multiplication: Multiplying a vector by a scalar (number) changes its magnitude but not its direction.
• Properties:
  • k(A + B) = kA + kB
  • (k + l)A = kA + lA
• Types of vector multiplication:
  • Dot product (scalar product): Produces a scalar value.
  • Cross product (vector product): Produces a vector value.

Cross Product (Vector Product)

• Definition: The cross product of two vectors A and B is a vector C that is perpendicular to both A and B.
• Notation: A × B or A ∧ B
• Properties:
  • A × B = - B × A (Anti-commutative)
  • A × (B + C) = A × B + A × C (Distributive)
• Right-hand rule: Use the right-hand rule to determine the direction of the cross product.

Dot Product (Scalar Product)

• Definition: The dot product of two vectors A and B is a scalar value that represents the amount of "similarity" between the two vectors.
• Notation: A · B or A ⋅ B
• Properties:
  • A · B = B · A (Commutative)
  • A · (B + C) = A · B + A · C (Distributive)
• Geometric interpretation: The dot product is related to the cosine of the angle between the two vectors.

Vector Projections

• Definition: The projection of a vector A onto a vector B is a vector that represents the component of A in the direction of B.
• Notation: projB(A)
• Formula: projB(A) = (A · B / ||B||²) B
• Properties:
  • projB(A) is parallel to B
  • ||projB(A)|| ≤ ||A||
• Applications: Finding the component of a force in a specific direction, resolving a force into its components.

Vector Projections

• The vector projection of b onto a is denoted as projₐ(b) or (b)ₐ, representing how much of b is in the direction of a.
• Formula: projₐ(b) = ((b · a) / (a · a)) a.

Dot Product

• The dot product is a scalar value that represents the amount of "similarity" between two vectors.
• Formula: a · b = a₁b₁ + a₂b₂ +...+ aₙbₙ.
• Properties:
  • a · b = b · a (commutative).
  • a · (b + c) = a · b + a · c (distributive).
  • a · a ≥ 0, with equality if and only if a = 0 (positive definite).

Cross Product

• The cross product is a vector value that represents the "perpendicularity" between two vectors.
• Formula: a × b = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁).
• Properties:
  • a × b = -(b × a) (anti-commutative).
  • a × (b + c) = a × b + a × c (distributive).
  • (a × b) × c ≠ a × (b × c) (not associative).

Vector Multiplication

• Scalar multiplication: k a = (ka₁, ka₂,..., kaₙ).
• Properties:
  • k(a + b) = k a + k b (distributive).
  • (k l) a = k(l a) (associative).

Vector Addition

• Adding two or more vectors by adding corresponding elements.
• Formula: a + b = (a₁ + b₁, a₂ + b₂,..., aₙ + bₙ).
• Properties:
  • a + b = b + a (commutative).
  • (a + b) + c = a + (b + c) (associative).
  • a + 0 = a (additive identity).
  • a + (-a) = 0 (additive inverse).