Vector Basics and Operations

Vector Basics

• A vector is a mathematical object with both magnitude (length) and direction.
• Vectors can be represented graphically as arrows in a coordinate plane.
• Vectors can be added and scaled (multiplied by a number).

Vector Operations

• Vector Addition: Two or more vectors can be added by combining their corresponding components.
  • Example: a = <1, 2>, b = <3, 4> then a + b = <1+3, 2+4> = <4, 6>
• Scalar Multiplication: A vector can be multiplied by a number (scalar) to change its magnitude.
  • Example: a = <1, 2>, k = 2 then ka = 2<1, 2> = <2, 4>
• Vector Subtraction: Subtracting one vector from another is equivalent to adding the opposite vector.
  • Example: a = <1, 2>, b = <3, 4> then a - b = a + (-b) = <1, 2> + <-3, -4> = <-2, -2>

Vector Properties

• Commutative Property: The order of vectors does not change the result of vector addition.
  • Example: a + b = b + a
• Associative Property: The order in which vectors are added does not change the result.
  • Example: (a + b) + c = a + (b + c)
• Distributive Property: Scalar multiplication can be distributed over vector addition.
  • Example: k(a + b) = ka + kb

Vector Magnitude and Direction

• Magnitude (Length): The length of a vector can be calculated using the Pythagorean theorem.
  • Example: a = <3, 4> then |a| = sqrt(3^2 + 4^2) = 5
• Direction: The direction of a vector can be described using angles or unit vectors.
  • Example: a = <3, 4> then the direction of a is tan^-1(4/3) or a/|a| = <3/5, 4/5>

Unit Vectors

• Unit Vectors: Vectors with a magnitude of 1, used to describe direction.
• i, j, k: Unit vectors in the x, y, and z directions, respectively.
  • Example: a = 3i + 4j then a is a vector with a magnitude of 5 in the direction of tan^-1(4/3)

Vector Basics

• A vector is a mathematical object with both magnitude (length) and direction.
• Vectors can be represented graphically as arrows in a coordinate plane.
• Vectors can be added and scaled (multiplied by a number).

Vector Operations

• Vectors can be added by combining their corresponding components.
• Scalar multiplication can change the magnitude of a vector by multiplying it by a number.
• Vector subtraction is equivalent to adding the opposite vector.

Vector Addition

• The order of vectors does not change the result of vector addition (Commutative Property).
• The order in which vectors are added does not change the result (Associative Property).

Scalar Multiplication

• Scalar multiplication can be distributed over vector addition (Distributive Property).

Vector Magnitude and Direction

• The magnitude (length) of a vector can be calculated using the Pythagorean theorem.
• The direction of a vector can be described using angles or unit vectors.

Unit Vectors

• Unit vectors have a magnitude of 1 and are used to describe direction.
• i, j, and k are unit vectors in the x, y, and z directions, respectively.
• Unit vectors can be used to describe the direction of a vector.