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Many physical entities (e.g., mass and energy) are
entirely defined by a numerical magnitude (expressed in appropriate units). Such entities, which have no directional element, are
known as scalars. Moreover, because scalars can be represented by real numbers,
it follows that they obey the laws of ordinary algebra. However, there exits a
second class of physical entities (e.g., velocity, acceleration, and force) that are
only completely defined when both a numerical magnitude and a direction in space are specified.
Such entities are known as vectors. By definition, a vector obeys the same algebra as
a displacement in space, and may thus be represented geometrically by a
straightline,
(say), where the arrow
indicates the direction of the displacement (i.e., from point
to point
). (See Figure A.1.)
The magnitude of the vector is represented by the length of the straightline.
Figure A.1:
A vector.

It is conventional to denote vectors by boldfaced symbols (e.g.,
,
) and
scalars by nonboldfaced symbols (e.g.,
,
). The magnitude of
a general vector,
, is denoted
, or just
, and is, by definition, always
greater than or equal to zero. It is convenient to define a vector with zero magnitudethis is
denoted
, and has no direction. Finally, two vectors,
and
, are said
to be equal when their magnitudes and directions are both identical.
Figure A.2:
Vector addition.

Next: Vector Algebra
Up: Vectors and Vector Fields
Previous: Introduction
Richard Fitzpatrick
20160331