Column and Row Vectors

Understanding Column and Row Vectors

A vector is a quantity that has both magnitude and direction. In its representation, vectors can be written in two forms: column vectors and row vectors.

Column vectors are vectors whose components are written vertically (downward). While row vectors are vectors whose components are written horizontally (sideways).

Notation of Column and Row Vectors

Here are the notations for column and row vectors:

1. For two-dimensional vectors:

   • Row vector: $(3 \; 4)$ or $[3 \; 4]$
   • Column vector: $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ or $\begin{bmatrix} 3 \\ 4 \end{bmatrix}$

2. For three-dimensional vectors:

   • Row vector: $(2 \; 1 \; 3)$ or $[2 \; 1 \; 3]$
   • Column vector: $\begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}$ or $\begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}$

Types of Column Vector Notation

In mathematics, column vectors can be written using various notations:

1. Notation with regular parentheses:

   $$\begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}$$

2. Notation with square brackets:

   $$\begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}$$

3. Notation with vertical bars:

   $$\begin{vmatrix} 2 \\ 1 \\ 3 \end{vmatrix}$$

All three notations represent the same column vector, only using different delimiters/brackets.

Unit Vectors in Cartesian Coordinate System

Unit vectors are vectors that have a length of 1 unit with a specific direction. In the Cartesian coordinate system, we recognize several unit vectors:

Unit Vectors in Two-Dimensional Cartesian Coordinate System

1. Unit vector in the horizontal direction (x-axis):

   • Row vector: $(1 \; 0)$
   • Column vector: $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$

2. Unit vector in the vertical direction (y-axis):

   • Row vector: $(0 \; 1)$
   • Column vector: $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$

Unit Vectors in Three-Dimensional Cartesian Coordinate System

1. Unit vector in the x-axis direction:

   • Row vector: $(1 \; 0 \; 0)$
   • Column vector: $\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$

2. Unit vector in the y-axis direction:

   • Row vector: $(0 \; 1 \; 0)$
   • Column vector: $\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$

3. Unit vector in the z-axis direction:

   • Row vector: $(0 \; 0 \; 1)$
   • Column vector: $\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$

Relationship Between Column and Row Vectors

Column and row vectors are related through the transpose operation. Transpose is an operation that changes rows into columns or vice versa.

If $\vec{a} = (a_1 \; a_2 \; \ldots \; a_n)$ is a row vector, then its transpose is a column vector:

Conversely, if $\vec{b} = \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{pmatrix}$ is a column vector, then its transpose is a row vector:

Applications of Column and Row Vectors

The distinction between column and row vector notation is very important in linear algebra operations such as matrix multiplication. In vector calculus, physics, and various scientific applications, the form of vector used must be consistent to ensure correct calculations.

In computing, column and row vectors are used to represent data and variables in numerical programming, image processing, artificial intelligence, and many other fields.