Calculus: Analytic Geometry and Calculus, with Vectors

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