Vectors and scalers

Scalars and Vectors

Scalars

• A scalar is a quantity with only magnitude (amount or size) but no direction.
• Examples: temperature, mass, time, energy, speed
• Scalars can be added or subtracted by simple arithmetic operations.
• Scalars can be multiplied or divided by numbers.

Vectors

• A vector is a quantity with both magnitude (amount or size) and direction.
• Examples: displacement, velocity, acceleration, force, momentum
• Vectors can be represented graphically using arrows in a coordinate system.
• Vectors have both magnitude (length of the arrow) and direction (direction of the arrow).

Vector Operations

• Addition: Vectors can be added by combining their corresponding components.
• Scalar Multiplication: A vector can be multiplied by a number, which changes its magnitude but not its direction.
• Vector Subtraction: Vectors can be subtracted by adding the negative of one vector to another.

Vector Notation

• Column Notation: Vectors can be represented as columns of numbers, e.g. |a| |b|
• Magnitude: The magnitude of a vector can be represented using the symbol | | or ||, e.g. |a|
• Unit Vectors: Vectors with a magnitude of 1, used to represent direction, e.g. i, j, k

Vector Properties

• Commutative Property: The order of vectors does not change the result of addition, e.g. a + b = b + a
• Associative Property: The order in which vectors are added does not change the result, e.g. (a + b) + c = a + (b + c)
• Distributive Property: A scalar can be distributed to each component of a vector, e.g. a(b + c) = ab + ac

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