Calculus: Analytic Geometry and Calculus, with Vectors

(lu) #1

68 Vectors and geometryin threedimensions


long run, more sensible to concentrate upon the two formulas

I(2.33)

Jul IvI cos 0
u.v _ U101 + UIVI+ u3t73

and to realize that the formula for cos 8 can be obtained very quickly by
equating the two formulas for
When, as above,
v=vli+vzj+rk

and lvi > 0, the vector in the parentheses in the right member of the
formula

(2.34) v = IVI (L,
i +Ivl j + Ivl

k)


is the unit vector in the direction of v. It is possible, and sometimes
thought to be useful, to recognize that the scalar components of the unit
vector are the cosines of the angles a, $, y which the vector v makes with
the unit vectors 1, j, k. This is true because the formulas

vi = IVI cos a, vj = IVI cos 0, vk = IVI cos y
vi=v1 vj =vg vk=v3

imply that

(2.341) ICI= cos a, cos 0, lol = cos y.

The angles a, 0, y, shown in Figure 2.342, are called the direction angles
of the vector v. The cosines of these angles are called the direction cosines
of the vector. Even those who do not like to prove formulas by use of
special figures in E3 should look at Figure 2.342 and see that the formulas

Figure 2.342

(2.341) can be proved by use of the formula that defines cosines in terms
of coordinates and distances. Squaring and adding the members of the
equations in (2.341) gives the formula

(2.343) cost a + cost 0 + toss y = I
Free download pdf