# 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 ($\stackrel{⃗}{v}$, $\stackrel{⃗}{u}$, 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: $\stackrel{⃗}{v}+\stackrel{⃗}{u}=\stackrel{⃗}{u}+\stackrel{⃗}{v}$
• Associative Property: $\stackrel{⃗}{v}+\left(\stackrel{⃗}{u}+\stackrel{⃗}{w}\right)=\left(\stackrel{⃗}{v}+\stackrel{⃗}{u}\right)+\stackrel{⃗}{w}$
• Distributive Property: $a\left(\stackrel{⃗}{u}+\stackrel{⃗}{v}\right)=a\stackrel{⃗}{u}+a\stackrel{⃗}{v}$
• 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 $\mathrm{\mid }\stackrel{⃗}{v}\mathrm{\mid }$or $\mathrm{\mid }\mathrm{\mid }\stackrel{⃗}{v}\mathrm{\mid }\mathrm{\mid }$.
• Unit Vector: A vector with a magnitude of 1 in the same direction as the original vector.

### Dot Product (Scalar Product):

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

### Cross Product (Vector Product):

• Definition: $\stackrel{⃗}{v}×\stackrel{⃗}{u}=\mathrm{\mid }\stackrel{⃗}{v}\mathrm{\mid }\cdot \mathrm{\mid }\stackrel{⃗}{u}\mathrm{\mid }\cdot \mathrm{sin}\left(\theta \right)\cdot \stackrel{^}{n}$, where $\stackrel{^}{n}$ is the unit normal vector perpendicular to the plane formed by $\stackrel{⃗}{v}$ and $\stackrel{⃗}{u}$.
• 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.