# Nakafa Framework: LLM

URL: /en/subject/high-school/10/mathematics/vector-operations/column-row-vector
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/mathematics/vector-operations/column-row-vector/en.mdx

Output docs content for large language models.

---

export const metadata = {
   title: "Column and Row Vectors",
   description: "Learn column and row vector notation, transpose operations, unit vectors in Cartesian systems, and their applications in linear algebra calculations.",
   authors: [{ name: "Nabil Akbarazzima Fatih" }],
   date: "04/12/2025",
   subject: "Vector and Operations",
};

## 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: <InlineMath math="(3 \; 4)" /> or <InlineMath math="[3 \; 4]" />
   - Column vector: <InlineMath math="\begin{pmatrix} 3 \\ 4 \end{pmatrix}" /> or <InlineMath math="\begin{bmatrix} 3 \\ 4 \end{bmatrix}" />

2. For three-dimensional vectors:
   - Row vector: <InlineMath math="(2 \; 1 \; 3)" /> or <InlineMath math="[2 \; 1 \; 3]" />
   - Column vector: <InlineMath math="\begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}" /> or <InlineMath math="\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:

   <BlockMath math="\begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}" />

2. Notation with square brackets:

   <BlockMath math="\begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}" />

3. Notation with vertical bars:

   <BlockMath math="\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: <InlineMath math="(1 \; 0)" />
   - Column vector: <InlineMath math="\begin{pmatrix} 1 \\ 0 \end{pmatrix}" />

2. Unit vector in the vertical direction (y-axis):
   - Row vector: <InlineMath math="(0 \; 1)" />
   - Column vector: <InlineMath math="\begin{pmatrix} 0 \\ 1 \end{pmatrix}" />

### Unit Vectors in Three-Dimensional Cartesian Coordinate System

1. Unit vector in the x-axis direction:

   - Row vector: <InlineMath math="(1 \; 0 \; 0)" />
   - Column vector: <InlineMath math="\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}" />

2. Unit vector in the y-axis direction:

   - Row vector: <InlineMath math="(0 \; 1 \; 0)" />
   - Column vector: <InlineMath math="\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}" />

3. Unit vector in the z-axis direction:
   - Row vector: <InlineMath math="(0 \; 0 \; 1)" />
   - Column vector: <InlineMath math="\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 <InlineMath math="\vec{a} = (a_1 \; a_2 \; \ldots \; a_n)" /> is a row vector, then its transpose is a column vector:

<BlockMath math="\vec{a}^T = \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix}" />

Conversely, if <InlineMath math="\vec{b} = \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{pmatrix}" /> is a column vector, then its transpose is a row vector:

<BlockMath math="\vec{b}^T = (b_1 \; b_2 \; \ldots \; b_m)" />

## 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.
