The dot product, also called the scalar product, of two vector s is a number ( scalar quantity) obtained by performing a specific operation on the vector components. The dot product has meaning only for pairs of vectors having the same number of dimensions. The symbol for dot product is a heavy dot ( ).

In the two-dimensional Cartesian plane, vectors are expressed in terms of the *x* -coordinates and *y* -coordinates of their end points, assuming they begin at the origin ( *x* , *y* ) = (0,0). Some examples are shown in the illustration below.

The dot product of two vectors is determined by multiplying their *x* -coordinates, then multiplying their *y* -coordinates, and finally adding the two products. Thus, in the above example:

**A** **B** = (2 x -4) + (5 x -3) = -8 - 15 = -23

**B** **C** = (-4 x 5) + (-3 x -5) = -20 + 15 = -5

**C** **A** = (5 x 2) + (-5 x 5) = 10 - 25 = -15

In polar coordinates, vectors are expressed in terms of length (magnitude) and direction. When expressed in this format, the dot product of two vectors is equal to the product of their lengths, multiplied by the cosine of the angle between them.

For any two vectors **A** and **B** , **A** **B** = **B** **A** . That is, the dot product operation is commutative; it does not matter in which order the operation is performed.

Also see Fast Reference for Math Symbols .

*This was last updated in*March 2011

*Posted by:*Margaret Rouse

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