2.2 Velocity
Like force, velocity is a vector with
direction and magnitude, with SI units of . Using the vector
to
denote the position of a particle of ﬂuid, its velocity is

(2.4) 
Vector ﬁelds
While can be used to denote a single velocity
with magnitude and direction, it can also denote a vector ﬁeld of velocity which varies
from point to point across a spatial domain. A vector is
represented by 3 numbers, relating to the coordinate system being
used, e.g. , , , in the Cartesian
system.
While the magnitude and direction of a vector is
ﬁxed, it is not invariant since the 3 values depend on the coordinate system
used. We represent a vector without reference to the coordinate
system by bold text, e.g.
“”
(compared to scalar “”).
Basic vector
algebra
Addition and subtraction of 2 vectors is performed by operating
on respective components. Subtraction of two vectors and
is
performed by
Multiplication
of any vector
by a scalar
is
performed by multiplying all the components by the scalar,
e.g.
Addition and multiplication are
commutative,
i.e. variables can be in
any order (
). Subtraction is not commutative. Products between scalars
and vectors are
distributive,
i.e.
and with additional scalar
,
Division
between a vector
and a scalar is
only relevant when the scalar is the second argument of the
operation,
i.e.