# Wolfram Language & System 10.0 (2014)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# Dot

or Dot[a,b,c]
gives products of vectors, matrices, and tensors.

## DetailsDetails

• gives an explicit result when a and b are lists with appropriate dimensions. It contracts the last index in a with the first index in b.
• Various applications of Dot:
•  {a1,a2}.{b1,b2} scalar product of vectors {a1,a2}.{{m11,m12},{m21,m22}} product of a vector and a matrix {{m11,m12},{m21,m22}}.{a1,a2} product of a matrix and a vector {{m11,m12},{m21,m22}}.{{n11,n12},{n21,n22}} product of two matrices
• The result of applying Dot to two tensors and is the tensor . Applying Dot to a rank tensor and a rank tensor gives a rank tensor. »
• Dot can be used on SparseArray objects, returning a SparseArray object when possible. »
• When its arguments are not lists or sparse arrays, Dot remains unevaluated. It has the attribute Flat.

## ExamplesExamplesopen allclose all

### Basic Examples  (3)Basic Examples  (3)

Scalar product of vectors:

 Out[1]=

Products of matrices and vectors:

 Out[1]=
 Out[2]=
 Out[3]=

Matrix product:

 Out[1]=

## TutorialsTutorials

Introduced in 1988
(1.0)
| Updated in 2003
(5.0)