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.

In two dimensions, the components form the row $(3 \; 4)$ or $[3 \; 4]$ . Written as a column, they keep the same order:

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

In three dimensions, the row $(2 \; 1 \; 3)$ or $[2 \; 1 \; 3]$ becomes the following column. Component order stays the same even though the arrangement changes.

\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:

Notation with regular parentheses:

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

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

For the standard Cartesian basis, we choose one unit vector in each positive axis direction. A unit vector in the negative direction also exists, but it is the opposite of the chosen basis 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.

Direction	Row vector
horizontal, along the $x$ -axis	$(1 \; 0)$
vertical, along the $y$ -axis

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

\begin{pmatrix} 1 \\ 0 \end{pmatrix} \qquad \begin{pmatrix} 0 \\ 1 \end{pmatrix}

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.

Direction	Row vector
along the $x$ -axis	$(1 \; 0 \; 0)$
along the -axis

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

\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \qquad \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \qquad \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}

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:

\vec{a}^T = \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix}

Conversely, consider the column vector:

\vec{b} = \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{pmatrix}

Its transpose is a row vector:

\vec{b}^T = (b_1 \; b_2 \; \ldots \; b_m)

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.

Row and column shapes determine the product shape. Take the two components already used above:

r=\begin{pmatrix}3&4\end{pmatrix},\qquad c=\begin{pmatrix}3\\4\end{pmatrix}

A row multiplied by a column produces one value because the result has shape $1\times1$ :

rc=3\cdot3+4\cdot4=9+16=25

A column multiplied by a row produces a $2\times2$ matrix. Each column entry multiplies both row entries:

cr=\begin{pmatrix}3\cdot3&3\cdot4\\4\cdot3&4\cdot4\end{pmatrix}=\begin{pmatrix}9&12\\12&16\end{pmatrix}

The component values are the same, but reversing the multiplication order gives objects with different shapes.