Column and Row Vectors

Writing Vector Components Vertically or Horizontally

A vector has magnitude and direction. In coordinate calculations, we commonly arrange its components as either a column vector or a row vector. A vector can also be shown with an arrow or other equivalent notation.

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

A column vector and a row vector carry the same components in the same order, so they differ only in how those components are laid out. Parentheses and square brackets are both accepted.

1. For two-dimensional vectors:

   • Row vector: $(3 \; 4)$ or $[3 \; 4]$

   • Column vector:

     $$\begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$$

2. For three-dimensional vectors:

   • Row vector: $(2 \; 1 \; 3)$ or $[2 \; 1 \; 3]$

   • Column vector:

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

Equivalent Component Representations

The same component vector can be represented in several useful ways:

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 Cartesian unit vectors:

   $$2\mathbf{i} + \mathbf{j} + 3\mathbf{k}$$

All three expressions describe the same vector. The first two display its components as a column, while the third expresses those components along the Cartesian unit vectors. Symbols with vertical bars must be read from context: $\det A$ or $|A|$ may denote the determinant of a square matrix, $|x|$ denotes the absolute value of a scalar, and $\lVert \vec{v} \rVert$ denotes the magnitude, or norm, of a vector.

Unit Vectors in Cartesian Coordinate System

Every coordinate axis carries exactly one basis vector, and those basis vectors express the coordinates of every other vector.

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

Unit Vectors in Two Dimensional Cartesian Coordinate System

The plane has two basis directions, and each basis vector has length one. The table gives both unit vectors in row form.

The same two unit vectors written as columns keep their components in the same order:

The first matrix is the horizontal unit vector, and the second matrix is the vertical unit vector.

Unit Vectors in Three Dimensional Cartesian Coordinate System

Space adds a third basis direction. The three unit vectors point along the three axes and each one has length one. The table gives them in row form.

The same three unit vectors written as columns keep their components in the same order:

The first matrix is the unit vector along the $x$-axis, the second is along the $y$-axis, and the third is along the $z$-axis.

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, consider the column vector:

Its transpose is a row vector:

Applications of Column and Row Vectors

Column and row vectors have different shapes in operations such as matrix multiplication. Keep the chosen form consistent so that matrix dimensions match.

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