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Vector mathematics is fundamental in representing physical quantities with both magnitude and direction, such as displacement, velocity, and force. Vectors are depicted as directed line segments in multidimensional space and can be manipulated through addition, subtraction, and scalar multiplication. Multiplicative operations include the dot and cross products, with vectors playing a key role in physics, engineering, computer graphics, and optimization algorithms like gradient descent.

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## Definition of Vectors

### Magnitude and Direction

Vectors are mathematical entities that have both magnitude and direction, distinguishing them from scalar quantities

### Physical Quantities Represented by Vectors

Vectors are crucial for representing physical quantities such as displacement, velocity, and force in a multidimensional space

### Representations of Vectors

Vectors can be represented as column vectors or in component form using unit vectors in the x and y directions

## Operations on Vectors

### Addition and Subtraction

Vectors can be added and subtracted by combining their components or chaining movements together

### Scalar Multiplication

Scalar multiplication alters the magnitude of a vector by multiplying each component by a scalar value

### Dot and Cross Products

Vectors can be multiplied using the dot product, which yields a scalar value, or the cross product, which results in a vector orthogonal to the original vectors

## Properties and Applications of Vectors

### Algebraic Properties

Vectors exhibit properties such as commutativity, associativity, and distributivity, making vector operations consistent and predictable

### Applications in Various Fields

Vectors have a wide range of applications in fields such as physics, engineering, computer graphics, and machine learning

### Vector Calculus

Vector calculus is used to describe the motion of objects in kinematics and is also essential in fields such as fluid dynamics and electromagnetism

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