Vector Operations in Math

Vector

• A vector is a quantity with both magnitude (length) and direction.
• It is often represented graphically as an arrow in a two-dimensional or three-dimensional space.
• Vectors can be added and scaled (multiplied by a number).

Vector Operations

• Addition: The sum of two vectors is obtained by adding corresponding components.
  • a = (a_x, a_y) and b = (b_x, b_y) => a + b = (a_x + b_x, a_y + b_y)
• Scalar Multiplication: A vector can be scaled by multiplying it by a number.
  • a = (a_x, a_y) and k is a scalar => ka = (ka_x, ka_y)

Types of Vectors

• Unit Vector: A vector with a magnitude of 1.
• Zero Vector: A vector with a magnitude of 0.
• Equal Vectors: Vectors with the same magnitude and direction.

Equilibrium

• Equilibrium: A state where the net force acting on an object is zero.
• Conditions for Equilibrium:
  • The sum of all forces acting on an object is zero (∑F = 0).
  • The sum of all torques acting on an object is zero (∑τ = 0).

Types of Equilibrium

• Translational Equilibrium: The net force acting on an object is zero.
• Rotational Equilibrium: The net torque acting on an object is zero.
• Stable Equilibrium: An object returns to its original position after a small disturbance.
• Unstable Equilibrium: An object moves away from its original position after a small disturbance.
• Neutral Equilibrium: An object remains in its new position after a small disturbance.

Vector

• A vector has both magnitude (length) and direction, and is often represented as an arrow in 2D or 3D space.
• Vectors can be added and scaled (multiplied by a number).

Vector Operations

• Vector addition is performed by adding corresponding components: (a_x, a_y) + (b_x, b_y) = (a_x + b_x, a_y + b_y).
• Scalar multiplication is performed by multiplying a vector by a number: k(a_x, a_y) = (ka_x, ka_y).

Types of Vectors

• A unit vector has a magnitude of 1.
• A zero vector has a magnitude of 0.
• Equal vectors have the same magnitude and direction.

Equilibrium

• Equilibrium is a state where the net force acting on an object is zero.
• The conditions for equilibrium are:
  • The sum of all forces acting on an object is zero (∑F = 0).
  • The sum of all torques acting on an object is zero (∑τ = 0).

Types of Equilibrium

• Translational equilibrium occurs when the net force acting on an object is zero.
• Rotational equilibrium occurs when the net torque acting on an object is zero.
• Stable equilibrium occurs when an object returns to its original position after a small disturbance.
• Unstable equilibrium occurs when an object moves away from its original position after a small disturbance.
• Neutral equilibrium occurs when an object remains in its new position after a small disturbance.