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1. A vector can be decomposed into a linear combination of the basis
vectors.
2. The dot product of two vectors can be written as the multiplication
of a row vector by a column vector, each of whose elements are
the components of the respective vectors.
3. The norm of a vector, a non-negative quantity, is the square root
of the dot product of the vector with itself.
4. The unit vector parallel to a specific vector is that vector divided
by its norm.
5. The projection of a vector on another can be deduced from the dot
product of the two vectors.

To facilitate the statement of these results in a notation that will be suitable
for infinite-dimensional vector spaces (which is very briefly introduced in
Section 7.7), Dirac in his elegant formulation of quantum mechanics intro-
duced a simple notation that we now present.
The Dirac notation represents the row vector by what he called the “bra-
vector” and the column vector by what he called the “ket-vector,” such that
when a dot product is obtained by joining the two vectors, the result will be
the scalar “bra-ket” quantity. Specifically:
r
Column vector u ⇒ u (7.23)
r
Row vector v ⇒ v (7.24)

r r
Dot product vu⋅⇒ v u (7.25)

The orthonormality of the basis vectors is written as:

mn =δ (7.26)
mn,
where the basis vectors are referred to by their indices, and where δ m,n is the
Kroenecker delta, equal to 1 when its indices are equal, and zero otherwise.
The norm of a vector, a non-negative quantity, is given by:

2
2
(norm of u ) = u = u u (7.27)
The Decomposition rule is written as:

u = ∑ c n (7.28)
n
n
where the components are obtained by multiplying Eq. (7.28) on the left by
m . Using Eq. (7.26), we deduce:

© 2001 by CRC Press LLC