Vectors in Math

Study Notes

Definition: A vector is a quantity with both magnitude (size) and direction.
Representation: Vectors are represented graphically as arrows in a coordinate system.
Operations:
- Addition: Vectors can be added by adding corresponding components.
- Scalar Multiplication: A vector can be multiplied by a number (scalar) to change its magnitude.
- Dot Product: The dot product of two vectors is a scalar value that represents the amount of "similarity" between the vectors.
Types:
- Displacement: A vector that represents the change in position of an object.
- Velocity: A vector that represents the rate of change of an object's position.
- Acceleration: A vector that represents the rate of change of an object's velocity.

Definition: A scalar is a quantity with only magnitude (size), no direction.
Examples:
- Temperature
- Time
- Mass
- Energy
Operations:
- Addition: Scalars can be added and subtracted.
- Multiplication: Scalars can be multiplied and divided.
Relationship with Vectors:
- Scalars can be used to scale vectors.
- Scalars can be used to describe the magnitude of a vector.

Magnitude: The size or length of a vector.
Direction: The direction in which a vector points.
Unit Vectors: Vectors with a magnitude of 1, used to specify direction.
Scalar Quantities: Quantities that can be described by a single number, such as temperature or mass.

A vector is a quantity with both magnitude (size) and direction, represented graphically as arrows in a coordinate system.
Vectors can be added by adding corresponding components, and multiplied by a number (scalar) to change their magnitude.
The dot product of two vectors is a scalar value representing the amount of "similarity" between the vectors.
Types of vectors include displacement (change in position), velocity (rate of change of position), and acceleration (rate of change of velocity).

A scalar is a quantity with only magnitude (size), no direction, and examples include temperature, time, mass, and energy.
Scalars can be added, subtracted, multiplied, and divided, and are used to scale vectors and describe their magnitude.

Magnitude refers to the size or length of a vector.
Direction refers to the direction in which a vector points.
Unit vectors have a magnitude of 1 and are used to specify direction.
Scalar quantities, such as temperature or mass, can be described by a single number.