Observe, however, that if P = (p 1 ,p 2 ,p 3 ) and Q =
(q 1 ,q 2 ,q 3 ) are two points on the line, then u =
→
PQ = – p 1 ,q 2 – p 2 ,q 3 – p 3 > is a VECTOR with direction along
the line, and any other point on the line can be found
by placing a scalar multiple of u at position P. That is,
the equation:
P + tu = (p 1 + t(q 1 – p 1 ), p 2 + t(q 2 – p 2 ), p 3 + t(q 3 – p 3 ))
= (1 – t)P + tQ
with t varying over all real numbers, describes all the
points on the line. This is called the vector equation of
the line. (Technically, one should work with the position vector p =
rather than the point P =
(p 1 ,p 2 ,p 3 ), so that the sum on the left of the above
equation is the addition of two like quantities.)
The formula y = mx + b, describing the equation of
a line in the plane, can also be thought of as a vector
equation. Here, points (x,y) on the line are given by:
(0,b) + x(1,m)
where (0,b) is one point of the line (the y-intercept) and
u = < 1,m > is the direction the line takes (1 step over
and m steps up).
vector equation of a plane The mathematical equation of a plane is a formula of the form:
ax + by + cz = d
where a, b, c, and d are numbers. Every point (x,y,z)
satisfying this equation is a point on a plane with VECTOR n = < a,b,c > as NORMAL TO THE PLANE.
To derive this formula, let P = (p 1 ,p 2 ,p 3 ) be a point
on a given plane and n = < a,b,c > a vector perpendicular to the plane. To find the equation for any other
point Q = (x,y,z) on this plane, note that the POSITION
VECTOR
→
PQ = < x – p 1 , y – p 2 , z – p 3 > lies in the plane
and is consequently perpendicular to n. The DOT PRODUCT n ·
→
PQ is thus zero. This gives the equation:
< a,b,c > · < x – p 1 ,y – p 2 ,z – p 3 >
= a(x – p 1 ) + b(y – p 2 ) + c(z – p 3 ) = 0
which can be rewritten:
ax + by + cz = d
where d is just a number.
Notice that the components of n appear as the
coefficients of the variables in the equation. This allows
one to quickly “read off” normal vectors to planes. For
example, the plane 2x – 3y + 4z = 10 has normal vector
n = < 2, –3, 4 >, as does the plane 2x – 3y + 4z = 6. The
plane y = 0 has normal vector n = < 0,1,0 >.
To find the equation of the plane with normal
vector n = < 2,2,1 >, say, and passing through the
point P = (5,–1,3), begin by writing the partial equation: 2x + 2y + z = d. To find d, substitute the coordinates of the point P. In this example we have: d = 2 . 5
+ 2·(–1) + 3 = 11, and so this plane has equation 2x +
2y + z = 11.
vector field A function that assigns to every point
in space a VECTOR is called a vector field. For example, wind currents on the Earth’s surface form a vector
field on the surface of the Earth: at every location
where there is a vector that describes the wind speed
and direction. In three-dimensional space there is a
gravitational vector field, given by the strength and
direction of gravitational force, that a unit mass
would experience when placed at each location in
space. The strength and direction of the force varies
from point to point.
The hairy ball theorem asserts that any smoothly
varying vector field across the surface of a sphere, with
vectors lying tangent to the sphere, must have at least
one vector that is the zero vector. It shows, for example,
that at any instant, there must be some location on the
Earth’s surface at which the horizontal wind speed is
zero. It also shows that it is impossible to comb all the
hairs of a tennis ball flat against the surface of the ball
in a smooth uniform fashion without ever producing a
cowlick. This second interpretation explains the name
of the theorem.
See also CURL; DIV.
vector space The set of two-dimensional or threedimensional VECTORs come equipped with two fundamental operations: vector addition and scalar
multiplication (that is, a multiplication by real numbers). These two operations obey the following rules:
vector space 521