Types of Vectors

Vectors are mathematical objects that have both magnitude and direction. They are used to represent quantities that have these two characteristics, such as force, velocity, and displacement. Vectors can be classified into several types based on their properties and the operations that can be performed on them. Below, we will explore the different types of vectors and their characteristics.

Zero Vector

A zero vector, also known as a null vector, is a vector with zero magnitude and an undefined direction. It is represented by the symbol $\vec{0}$.

• Magnitude: 0
• Direction: Undefined
• Notation: $\vec{0}$
• Example: $\vec{0} = \begin{pmatrix} 0 \ 0 \ 0 \end{pmatrix}$ in three-dimensional space

Unit Vector

A unit vector is a vector with a magnitude of 1. It is often used to specify a direction and is denoted by a hat symbol (ˆ) over the vector.

• Magnitude: 1
• Direction: Defined by the vector it is normalizing
• Notation: $\hat{v}$
• Example: $\hat{i} = \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix}$ is the unit vector along the x-axis in three-dimensional space

Position Vector

A position vector represents the position of a point in space relative to an origin. It is defined as the vector from the origin to the point.

• Magnitude: Varies depending on the point's position
• Direction: From the origin to the point
• Notation: $\vec{r}$
• Example: $\vec{r} = \begin{pmatrix} x \ y \ z \end{pmatrix}$ represents the position of point $(x, y, z)$ in three-dimensional space

Equal Vectors

Equal vectors have the same magnitude and direction. They are equivalent regardless of their initial points.

• Magnitude: Equal
• Direction: Equal
• Notation: $\vec{a} = \vec{b}$
• Example: If $\vec{a} = \begin{pmatrix} 3 \ 2 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} 3 \ 2 \end{pmatrix}$, then $\vec{a} = \vec{b}$

Parallel Vectors

Parallel vectors have the same or opposite direction but can have different magnitudes. They are scalar multiples of each other.

• Magnitude: Can vary
• Direction: Same or opposite
• Notation: $\vec{a} \parallel \vec{b}$
• Example: If $\vec{a} = \begin{pmatrix} 2 \ 4 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} 4 \ 8 \end{pmatrix}$, then $\vec{a} \parallel \vec{b}$ because $\vec{b} = 2\vec{a}$

Antiparallel Vectors

Antiparallel vectors are a special case of parallel vectors where they have opposite directions.

• Magnitude: Can vary
• Direction: Opposite
• Notation: $\vec{a} \uparrow\downarrow \vec{b}$
• Example: If $\vec{a} = \begin{pmatrix} 3 \ -2 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} -6 \ 4 \end{pmatrix}$, then $\vec{a} \uparrow\downarrow \vec{b}$ because $\vec{b} = -2\vec{a}$

Orthogonal Vectors

Orthogonal vectors are perpendicular to each other. Their dot product is zero.

• Magnitude: Can vary
• Direction: Perpendicular
• Notation: $\vec{a} \perp \vec{b}$
• Example: If $\vec{a} = \begin{pmatrix} 2 \ 3 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} -3 \ 2 \end{pmatrix}$, then $\vec{a} \perp \vec{b}$ because $\vec{a} \cdot \vec{b} = 0$

Coplanar Vectors

Coplanar vectors lie in the same plane. This is a concept that applies to three or more vectors.

• Magnitude: Can vary
• Direction: Can vary
• Notation: Not specifically denoted
• Example: Any three vectors that lie in the xy-plane are coplanar

Collinear Vectors

Collinear vectors lie along the same line. They are either parallel or antiparallel.

• Magnitude: Can vary
• Direction: Same line
• Notation: Not specifically denoted
• Example: If $\vec{a} = \begin{pmatrix} 1 \ 2 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} 2 \ 4 \end{pmatrix}$, then $\vec{a}$ and $\vec{b}$ are collinear

Differences and Important Points

Type of Vector	Magnitude	Direction	Notation	Example
Zero Vector	0	Undefined	$\vec{0}$	$\vec{0} = \begin{pmatrix} 0 \ 0 \ 0 \end{pmatrix}$
Unit Vector	1	Defined by the vector it is normalizing	$\hat{v}$	$\hat{i} = \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix}$
Position Vector	Varies	From the origin to the point	$\vec{r}$	$\vec{r} = \begin{pmatrix} x \ y \ z \end{pmatrix}$
Equal Vectors	Equal	Equal	$\vec{a} = \vec{b}$	$\vec{a} = \vec{b}$ if $\vec{a} = \begin{pmatrix} 3 \ 2 \end{pmatrix}$
Parallel Vectors	Can vary	Same or opposite	$\vec{a} \parallel \vec{b}$	$\vec{a} \parallel \vec{b}$ if $\vec{a} = \begin{pmatrix} 2 \ 4 \end{pmatrix}$
Antiparallel Vectors	Can vary	Opposite	$\vec{a} \uparrow\downarrow \vec{b}$	$\vec{a} \uparrow\downarrow \vec{b}$ if $\vec{a} = \begin{pmatrix} 3 \ -2 \end{pmatrix}$
Orthogonal Vectors	Can vary	Perpendicular	$\vec{a} \perp \vec{b}$	$\vec{a} \perp \vec{b}$ if $\vec{a} = \begin{pmatrix} 2 \ 3 \end{pmatrix}$
Coplanar Vectors	Can vary	Can vary	Not specifically denoted	Any three vectors in the xy-plane
Collinear Vectors	Can vary	Same line	Not specifically denoted	$\vec{a}$ and $\vec{b}$ are collinear if $\vec{a} = \begin{pmatrix} 1 \ 2 \end{pmatrix}$

Understanding these types of vectors is crucial for solving problems in physics and engineering, as well as in various fields of mathematics such as vector calculus and linear algebra. Each type of vector has unique properties that can be exploited to simplify calculations and to understand geometric and physical relationships.