Vector Basics

What is a Vector?

• A vector is a quantity with both magnitude (length) and direction.
• It is often represented graphically as an arrow in a coordinate system.
• Vectors can be added and scaled (multiplied by a number).

Vector Notation

• Vectors are typically denoted by boldface letters (e.g. a, b, c).
• The magnitude of a vector is denoted by | | or || || (e.g. |a| or ||a||).
• The direction of a vector is denoted by the unit vector (e.g. a/|a|).

Operations with Vectors

Vector Addition

• The sum of two vectors is a vector.
• The resultant vector is found by adding corresponding components.
• The order of addition does not change the result.

Scalar Multiplication

• Multiplying a vector by a number (scalar) changes its magnitude but not its direction.
• The resulting vector is parallel to the original vector.

Vector Properties

Commutative Property

• The order of vectors in addition does not change the result.

Associative Property

• The order of vectors in addition can be rearranged without changing the result.

Distributive Property

• Scalar multiplication can be distributed over vector addition.

Types of Vectors

Unit Vectors

• A unit vector has a magnitude of 1.
• It is used to denote direction.

Zero Vector

• A zero vector has a magnitude of 0.
• It is the additive identity.

Equal Vectors

• Two vectors are equal if they have the same magnitude and direction.

Vector Applications

• Physics: velocity, acceleration, force, momentum
• Engineering: design, stress, load
• Computer Science: graphics, game development, machine learning

What is a Vector?

• A vector is a quantity with both magnitude (length) and direction.
• It is often represented graphically as an arrow in a coordinate system.
• Vectors can be added and scaled (multiplied by a number).

Vector Notation

• Vectors are typically denoted by boldface letters (e.g. a, b, c).
• The magnitude of a vector is denoted by | | or || || (e.g. |a| or ||a||).
• The direction of a vector is denoted by the unit vector (e.g. a/|a|).

Operations with Vectors

Vector Addition

• The sum of two vectors is a vector.
• The resultant vector is found by adding corresponding components.
• The order of addition does not change the result.

Scalar Multiplication

• Multiplying a vector by a number (scalar) changes its magnitude but not its direction.
• The resulting vector is parallel to the original vector.

Vector Properties

Commutative Property

• The order of vectors in addition does not change the result.

Associative Property

• The order of vectors in addition can be rearranged without changing the result.

Distributive Property

• Scalar multiplication can be distributed over vector addition.

Types of Vectors

Unit Vectors

• A unit vector has a magnitude of 1.
• It is used to denote direction.

Zero Vector

• A zero vector has a magnitude of 0.
• It is the additive identity.

Equal Vectors

• Two vectors are equal if they have the same magnitude and direction.

Vector Applications

• Physics: used to describe velocity, acceleration, force, and momentum.
• Engineering: used in design, stress, and load calculations.
• Computer Science: used in graphics, game development, and machine learning.

Kinetic Energy

• Kinetic energy is the energy of motion, which an object possesses due to its motion.
• The formula to calculate kinetic energy (KE) is: KE = 1/2 × m × v^2
• In this formula, KE is measured in joules (J), m is the mass of the object in kilograms (kg), and v is the velocity of the object in meters per second (m/s).

Key Characteristics

• Kinetic energy depends on the square of the velocity, resulting in a significant increase in kinetic energy with a small increase in velocity.
• Kinetic energy is a scalar quantity, meaning it has no direction.
• The unit of kinetic energy is the joule (J), which is the same unit used to measure work and energy.

Examples of Kinetic Energy

• A rolling ball has kinetic energy due to its motion.
• A moving car has kinetic energy due to its velocity.
• A flying airplane has kinetic energy due to its motion through the air.

Conversion to Other Forms of Energy

• Kinetic energy can be converted to potential energy, such as when an object is lifted against gravity.
• Kinetic energy can be converted to thermal energy, such as when an object experiences friction and generates heat.
• Kinetic energy can be converted to electrical energy, such as when a generator converts kinetic energy into electrical energy.