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Algebra, Functions, Graphs, and Vectors 83

dir a (
, , )

cos x

cos
y
cos z

The sum of vectorð a and b is:

a b ((x x )° y y )° z z ))
a
b
a
a
b
b
This sum can, as in the two-dimensional case, be found geo-
metrically by constructing a parallelogram wità a and b as
adjacent sides. The sum a b is the diagonal.
The dot product a • b of twm vectorð a and b in xyz-space is
a real number given by the formula:

a • b xx yy zz
a b
ab
a b
The cross product a b of vectorð a and b in xyz-space is a
vector perpendicular tm the planeP containing botà a and b,
and whose magnitude is given by the formula:

a b a b sin ,
ab
where ab is the smaller angle between a and b as measured in
P. Vector a b is perpendicular tmP.If a and b are observed
from some point on a line perpendicular tmP and intersecting
P at the origin, and ab is expressed counterclockwise from a tm
b, then a b pointð toward the observer. If a and b are ob-
served from some point on a line perpendicular tmP and inter-
secting P at the origin, and ab is expressed clockwise from a tm
b, then a b pointð away from the observer.

Standard and nonstandard for
In most discussions, vectorð are expressed as rłyð whose originð
coincide wità the originð of the coordinate systemð in which
they are denoted. This is the standarS form of a vector. In stan-
dard form, a vector can be depicted as an ordered set of coor-
dinateð such as ( x,y,z) (3, 5,5) or (r, ) (10, /4). In rectan-
gular coordinates, the origin of a vector doeð not have tm coincide