of the vector is given by P
YTHAGORAS
’
S THEOREM
: |a| =
, which is just the
DISTANCE FORMULA
in this
new setting.
Vectors drawn in the plane are called two-dimen-
sional. Three-dimensional vectors, given by triples of
numbers, represent quantities with magnitude and
direction in three-dimensional space. S
IR
W
ILLIAM
R
OWAN
H
AMILTON
(1805–65) coined the term vector
for these quantities in his book Lectures on Quater-
nions. The name is based on the Latin word vectus, the
perfect participle of vehere, which means “to carry.”
Operations on Vectors
Vector addition combines two (or more) vectors to pro-
duce a new vector, called the resultant vector. To add
together two vectors aand b, position vector b(with-
out changing its magnitude or direction) so that its ini-
tial point lies at the terminal point of a. The resultant a
+ bis the vector that “carries particles” from the initial
point of ato the terminal point of b. Geometrically,
this is the vector that corresponds to the hypotenuse of
the triangle formed by aand b.
Algebraically, if a= <a1,a2> and b= <b1, b2>, then
a+ b= <a1+ b1, a2+ b2>.(If a“shifts particles” a1units
to the right and bshifts particles b1units to the right,
then the combined effect a+bshifts particles a1+ b1
rightward. Similarly for the combined vertical motion.)
From either viewpoint we see that vector addition is
COMMUTATIVE
:
a+ b= b+ a
One can also check that vector addition is
ASSOCIATIVE
:
a+ (b+ c) = (a+ b) + c
and that the zero vector acts as an identity element: a+
0= a= 0+ a.
Scalar multiplication multiplies vectors by numbers
to produce new vectors. In vector analysis it is custom-
ary to refer to numbers as scalars. If ais a vector and r
a scalar, then rais the vector with:
• The same direction as aif r> 0
• The opposite direction as aif r< 0
• Length |ra| equal to |r| |a|
Algebraically, if a= <a1,a2>, then ra= <ra1ra2>. Thus,
for example, if a= <6,–8>, then a= <3,–4> is the
vector half as long, pointing in the same direction;
–a= <–6,8> is the vector of the same length but
pointing in the reverse direction; and 0a= <0,0> = 0
is the zero vector.
Vector subtraction is performed by adding one vec-
tor to the negative of the other. For example, if aand b
are vectors, then the vector a– bis the result of adding
–bto a. Geometrically, this corresponds to reversing
the direction of the vector band placing this reversed
vector with initial point at the terminal point of a, as
illustrated below. We have: a+ (–a) = 0.
The operations of vector addition and scalar multi-
plication satisfy a number of basic rules, yielding a
mathematical system called a
VECTOR SPACE
.
There are two standard types of multiplication on
vectors. These are the
DOT PRODUCT
and the
CROSS
PRODUCT
.
A vector of length 1 is called a unit vector. In two-
dimensional space there are two standard unit vectors
i= <1,0> and j= <0,1>. Any two-dimensional vector a
= < a1,a2> can be written as a (linear) combination of
these two unit vectors:
a= < a1,a2> = a1<1,0> +a2< 0,1 > = a1i+a2j
Similarly, in three-dimensional space, there are three
standard unit vectors: i= <1,0,0>, j= <0,1,0>, and k=
<0,0,1>. These too form a
BASIS
for the space of three-
dimensional vectors. Physicists often prefer to represent
vectors as combinations of the standard unit vectors.
See also
ORTHOGONAL
;
PARALLELOGRAM LAW
;
POSITION VECTOR
;
PROJECTION
;
TENSOR
;
TRIPLE VEC
-
TOR PRODUCT
;
VECTOR EQUATION OF A LINE
;
VECTOR
EQUATION OF A PLANE
;
VECTOR FIELD
;
VECTOR SPACE
.
vector equation of a line The equation of a
LINE
in
two-dimensional space can be written in the form:
1
–
2
√a12+ a22
520 vector equation of a line
Vector operations