# Vectors

### Basics of Vectors:

#### Definition:

**Magnitude and Direction:**Vectors are quantities defined by both magnitude (size) and direction.**Representation:**Typically represented as arrows in space, with length indicating magnitude and direction showing its orientation.

#### Notation:

**Representation:**Vectors are denoted by symbols with an arrow on top ($\stackrel{\u20d7}{v}$, etc.).**Components:**Vectors can be represented by their components in various dimensions (2D, 3D, etc.).

#### Types of Vectors:

**Position Vectors:**Represent the position of a point in space relative to a reference point or origin.**Displacement Vectors:**Represent the change in position from one point to another.**Force, Velocity, Acceleration:**Common physical vectors representing force, speed with direction, and change in velocity, respectively.

### Vector Operations:

#### Addition and Subtraction:

**Graphical Method:**Vectors are added or subtracted by placing them tip-to-tail or by using the parallelogram law.**Component Method:**Add or subtract corresponding components of vectors.

#### Scalar Multiplication:

**Scaling:**Multiplying a vector by a scalar changes its magnitude without altering its direction.

#### Vector Properties:

**Commutative Property:**$\stackrel{\u20d7}{v}+\stackrel{\u20d7}{u}=\stackrel{\u20d7}{u}+\stackrel{\u20d7}{v}$- $\stackrel{\u20d7}{v}+(\stackrel{\u20d7}{u}+\stackrel{\u20d7}{w})=(\stackrel{\u20d7}{v}+\stackrel{\u20d7}{u})+\stackrel{\u20d7}{w}$
- $a(\stackrel{\u20d7}{u}+\stackrel{\u20d7}{v})=a\stackrel{\u20d7}{u}+a\stackrel{\u20d7}{v}$
**Magnitude:**The length of the vector, often represented as $\mathrm{\mid}\mathbf{v}\mathrm{\mid}$.**Unit Vector:**A vector with a magnitude of 1 in a specific direction.**Zero Vector:**A vector with zero magnitude; all its components are zero.**Parallel Vectors:**Have the same or opposite directions, even if their magnitudes differ.**Orthogonal (Perpendicular) Vectors:**Their dot product is zero.

### Magnitude and Direction:

**Magnitude:**The length or size of a vector, denoted by $\mathrm{\mid}\stackrel{\u20d7}{v}\mathrm{\mid}$.**Unit Vector:**A vector with a magnitude of 1 in the same direction as the original vector.

### Dot Product (Scalar Product):

**Definition:**- $\stackrel{\u20d7}{v}\cdot \stackrel{\u20d7}{u}=\mathrm{\mid}\stackrel{\u20d7}{v}\mathrm{\mid}\cdot \mathrm{\mid}\stackrel{\u20d7}{u}\mathrm{\mid}\cdot \mathrm{cos}(\theta )$, where $\theta $ is the angle between the vectors.
**Properties:**Commutative, distributive, and follows trigonometric rules.

### Cross Product (Vector Product):

**Definition:**$\stackrel{\u20d7}{v}\times \stackrel{\u20d7}{u}=\mathrm{\mid}\stackrel{\u20d7}{v}\mathrm{\mid}\cdot \mathrm{\mid}\stackrel{\u20d7}{u}\mathrm{\mid}\cdot \mathrm{sin}(\theta )\cdot \widehat{n}$.**Result:**A vector perpendicular to both original vectors.

### Applications:

**Physics:**Forces, moments, torque, and electromagnetism.**Engineering:**Structural analysis, mechanical systems.**Computer Graphics:**Manipulating objects in 3D space, lighting effects.