Math Vector Basics

Study Notes

A vector is a quantity with both magnitude (amount of movement) and direction.
Vectors are often represented graphically using arrows in a coordinate system.
Operations with vectors:
- Addition: combining two or more vectors by adding their corresponding components.
- Scalar multiplication: multiplying a vector by a number ( scalar) changes its magnitude but not its direction.
Types of vectors:
- Displacement vector: represents the change in position of an object.
- Velocity vector: represents the rate of change of an object's position.
- Acceleration vector: represents the rate of change of an object's velocity.

Work is the transfer of energy from one object to another through a force applied over a distance.
Formula: W = F × d × cos(θ), where:
- W is the work done (in joules, J)
- F is the force applied (in newtons, N)
- d is the distance over which the force is applied (in meters, m)
- θ is the angle between the force and the displacement vectors
Units: joules (J) or newton-meters (N·m)
Types of work:
- Positive work: force and displacement are in the same direction.
- Negative work: force and displacement are in opposite directions.
- Zero work: force and displacement are perpendicular.

Power is the rate at which work is done or energy is transferred.
Formula: P = W / t, where:
- P is the power (in watts, W)
- W is the work done (in joules, J)
- t is the time over which the work is done (in seconds, s)
Units: watts (W) or joules per second (J/s)
Types of power:
- Mechanical power: relates to the rate of doing mechanical work.
- Electrical power: relates to the rate of electrical energy transfer.

Energy is the ability to do work or cause change.
Types of energy:
- Kinetic energy: the energy of motion.
- Potential energy: the energy of position or stored energy.
- Thermal energy: the energy of heat.
- Electrical energy: the energy associated with the movement of charged particles.
Energy units:
- Joules (J)
- Kilowatt-hours (kWh) (1 kWh = 3.6 MJ)
Energy conservation: the total energy of an isolated system remains constant, but can be converted from one form to another.

A vector is a quantity with both magnitude and direction.
Graphical representation: arrows in a coordinate system.
Operations:
- Vector addition: combining vectors by adding corresponding components.
- Scalar multiplication: multiplying a vector by a number changes its magnitude, not direction.
Types of vectors:
- Displacement vector: represents change in position of an object.
- Velocity vector: represents rate of change of an object's position.
- Acceleration vector: represents rate of change of an object's velocity.

Work is the transfer of energy through a force applied over a distance.
Formula: W = F × d × cos(θ), where W is work done, F is force, d is distance, and θ is the angle between force and displacement vectors.
Units: joules (J) or newton-meters (N·m).
Types of work:
- Positive work: force and displacement are in the same direction.
- Negative work: force and displacement are in opposite directions.
- Zero work: force and displacement are perpendicular.

Power is the rate at which work is done or energy is transferred.
Formula: P = W / t, where P is power, W is work done, and t is time.
Units: watts (W) or joules per second (J/s).
Types of power:
- Mechanical power: relates to rate of doing mechanical work.
- Electrical power: relates to rate of electrical energy transfer.

Energy is the ability to do work or cause change.
Types of energy:
- Kinetic energy: energy of motion.
- Potential energy: energy of position or stored energy.
- Thermal energy: energy of heat.
- Electrical energy: energy associated with movement of charged particles.
Energy units:
- Joules (J)
- Kilowatt-hours (kWh) (1 kWh = 3.6 MJ)
Energy conservation: total energy of an isolated system remains constant, but can be converted from one form to another.