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Understanding scalar and vector quantities is fundamental in physics and geometry. Scalars have magnitude only, like temperature and mass, while vectors have both magnitude and direction, such as displacement and force. This text delves into graphical vector representation in planes, calculating vector magnitude and direction, vector operations like addition and subtraction, and their applications in solving geometric and physical problems.

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## Scalar and Vector Quantities

### Definition of Scalar Quantities

Scalar quantities are numerical values representing size or amount without any associated direction

### Definition of Vector Quantities

Vector quantities are characterized by both magnitude and direction

### Examples of Scalar and Vector Quantities

Temperature, mass, and distance are examples of scalar quantities, while displacement, force, and acceleration are examples of vector quantities

## Representing Vectors in a Two-Dimensional Plane

### Coordinate System

A coordinate system, such as the Cartesian xy-plane, is used to specify the direction and magnitude of vectors in a two-dimensional plane

### Geometric Representation of Vectors

Vectors in a two-dimensional plane can be depicted as arrows with their length proportional to their magnitude

### Mathematical Representation of Vectors

Vectors in a two-dimensional plane can be expressed as ordered pairs or in terms of unit vectors

## Three-Dimensional Vectors

### Additional Component in Three Dimensions

Three-dimensional vectors include an additional component along the z-axis, perpendicular to the x and y axes

### Representation of Three-Dimensional Vectors

Three-dimensional vectors can be represented by ordered triplets or in terms of unit vectors

### Complete Description of Orientation in Space

Three-dimensional vectors provide a complete description of their orientation in space

## Operations with Vectors

### Magnitude of Vectors

The magnitude of a vector, which is a measure of its length, can be calculated using the Pythagorean theorem

### Direction of Vectors

The direction of a vector is typically described by the angle it makes with a reference axis or line

### Vector Addition and Subtraction

Vector addition and subtraction involve combining vectors to produce a new vector, either geometrically or algebraically

### Multiplying Vectors by a Scalar

Multiplying a vector by a scalar changes its magnitude but not its direction

### Determining Distance between Points

The distance between two points represented by position vectors can be found by subtracting one vector from the other and calculating the magnitude of the resulting displacement vector

Algorino

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