Vectors in Physics

Definition of a Vector

• A vector is a quantity with both magnitude (length) and direction
• It is often represented graphically as an arrow in a coordinate plane
• Vectors are used to represent physical quantities with both size and direction, such as displacement, velocity, acceleration, and force

Types of Vectors

• Scalar: a quantity with only magnitude, no direction
• Vector: a quantity with both magnitude and direction
• Unit Vector: a vector with a magnitude of 1, used to specify direction
• Zero Vector: a vector with a magnitude of 0, used to represent no movement or force

Vector Operations

• Vector Addition: the sum of two or more vectors, resulting in a new vector
  • Commutative property: a + b = b + a
  • Associative property: (a + b) + c = a + (b + c)
• Scalar Multiplication: multiplying a vector by a scalar (number), resulting in a new vector
  • k * a = ka (where k is a scalar and a is a vector)
• Vector Subtraction: the difference between two vectors, resulting in a new vector
  • a - b = a + (-b)

Vector Properties

• Magnitude (Length): the size or length of a vector, often represented as |a|
• Direction: the direction in which a vector points, often represented as θ (theta)
• Unit Vector: a vector with a magnitude of 1, used to specify direction
• Orthogonality: two vectors are orthogonal (perpendicular) if their dot product is 0
• Parallel: two vectors are parallel if one is a scalar multiple of the other

Vector Applications

• Physics: vectors are used to represent displacement, velocity, acceleration, and force
• Engineering: vectors are used to represent stresses, strains, and motion in structures and machines
• Computer Science: vectors are used in computer graphics, game development, and machine learning algorithms

Definition of a Vector

• A vector is a quantity with both magnitude (length) and direction.
• Graphical representation: an arrow in a coordinate plane.
• Used to represent physical quantities with size and direction, such as displacement, velocity, acceleration, and force.

Types of Vectors

• Scalar: quantity with only magnitude, no direction.
• Vector: quantity with both magnitude and direction.
• Unit Vector: vector with a magnitude of 1, used to specify direction.
• Zero Vector: vector with a magnitude of 0, represents no movement or force.

Vector Operations

• Vector Addition: sum of two or more vectors, resulting in a new vector.
• Properties of vector addition: commutative (a + b = b + a) and associative ((a + b) + c = a + (b + c)).
• Scalar Multiplication: multiplying a vector by a scalar (number), resulting in a new vector.
• Vector Subtraction: difference between two vectors, resulting in a new vector.
• Subtraction property: a - b = a + (-b).

Vector Properties

• Magnitude (Length): size or length of a vector, often represented as |a|.
• Direction: direction in which a vector points, often represented as θ (theta).
• Unit Vector: vector with a magnitude of 1, used to specify direction.
• Orthogonality: two vectors are orthogonal (perpendicular) if their dot product is 0.
• Parallel: two vectors are parallel if one is a scalar multiple of the other.

Vector Applications

• Physics: vectors represent displacement, velocity, acceleration, and force.
• Engineering: vectors represent stresses, strains, and motion in structures and machines.
• Computer Science: vectors used in computer graphics, game development, and machine learning algorithms.