For AI agents: use /llms.txt for the Nakafa content index.
Column vectors are vectors whose components are written vertically (downward). While row vectors are vectors whose components are written horizontally (sideways).
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 or . Written as a column, they keep the same order:
In three dimensions, the row or becomes the following column. Component order stays the same even though the arrangement changes.
The same component vector can be represented in several useful ways:
Notation with regular parentheses:
Notation with square brackets:
Notation with Cartesian unit vectors:
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: or may denote the determinant of a square matrix, denotes the absolute value of a scalar, and denotes the magnitude, or norm, of a vector.
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 with a specific direction. In the Cartesian coordinate system, we recognize several unit vectors:
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 -axis | |
| vertical, along the -axis |
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.
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 -axis | |
| along the -axis | |
| along the -axis |
The same three unit vectors written as columns keep their components in the same order:
The first matrix is the unit vector along the -axis, the second is along the -axis, and the third is along the -axis.
Column and row vectors are related through the transpose operation. Transpose is an operation that changes rows into columns or vice versa.
If is a row vector, then its transpose is a column vector:
Conversely, consider the column vector:
Its transpose is a row vector:
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:
A row multiplied by a column produces one value because the result has shape :
A column multiplied by a row produces a matrix. Each column entry multiplies both row entries:
The component values are the same, but reversing the multiplication order gives objects with different shapes.
Published: . Updated: .
