A vector might be denoted by listing all its elements as
. This format is called a *row vector*. Alternatively,
the elements could be arranged as

which is a

When a vector is multiplied by a scalar (a number), this is the same as
multiplying each component of the vector by that scalar, e.g.,

(1) |

*For real numbers*, the dot (or inner) product between two vectors
and is given by

(2) |

If and are complex, then one small complication arises.
If and are given by

(3) |

then to take the dot product of and , we must first take the

(4) |

Recall that the complex conjugate of a complex number like is (the imaginary part has its sign reversed). Any time a row vector is converted into a column vector (or vice versa), if it is complex, then one needs to take complex conjugates of each element.

(5) |

A vector is said to be *normalized* if its dot product
with itself,
, is 1. If this is not the case, it is
always possible to normalize the vector to force the dot product to
come out to 1; one merely needs to divide each component of the vector by
the square root of the dot product that results before normalization.
For example, the vector

(6) |

(7) |

(8) |