and a triangle would not exist. By equating to zero the scalar coefficients in these
equations, there results the simultaneous scalar equations
(1 −−m 2 ) = 0 , (m 2 − 2 ) = 0 , (k 2 −n 2 ) = 0, (1 −n−k 2 ) = 0
The solution of these equations produces the fact that k==m=n=
2
3 and hence
the conclusion P=P∗ is a trisection point.
Unit Vectors
A vector having length or magnitude of one is called a unit vector. If A is
a nonzero vector of length |A|, a unit vector in the direction of A is obtained by
multiplying the vector A by the scalar m= |A^1 |. The unit vector so constructed is
denoted
ˆeA=
A
|A|
and satisfies |ˆeA|= 1.
The symbol ˆeis reserved for unit vectors and the notation ˆeAis to be read “a unit
vector in the direction of A.” The hat or carat (̂)notation is used to represent
aunit vector or normalized vector.
Figure 6-5. Cartesian axes.
The figure 6-5 illustrates unit base vectors eˆ 1 , eˆ 2 , ˆe 3 in the directions of the pos-
itive x, y, z -coordinate axes in a rectangular three dimensional cartesian coordinate
system. These unit base vectors in the direction of the x, y, z axes have historically
