Calculus: Analytic Geometry and Calculus, with Vectors

2.3 Scalar products, direction cosines, and lines in E3 71

point-direction equations of lines because they reveal coordinates of
points and scalar components of vectors lying on the lines.

Problems 2.39

1 Write the intrinsic (not depending upon coordinate system) formula for

the scalar product of two vectors u and v and be prepared to rewrite it at any

time.

2 Write the coordinate-dependent formula for the scalar product of the

two vectors

u = ail+b,j+c,k, v = asi+bsj+csk

and be prepared to rewrite it at any time.

3 Use the results of the first two problems to obtain a formula for the cosine

of the angle between the two vectors in Problem 2 and be prepared to repeat the

process at any time.

4 Find the scalar product of the two given vectors and use it to find cos 0,

the cosine of the angle between the vectors:

(a) u=2f-3j+ 4k, v = 21 + 3j + 4k 4nr.:cos0=

(b)u=21-3j-4k,v=21+3j+4k .Ins.:cos0=^29

(c) u = 21, + 3j + 4k, v = 21 + 3j + 4k .Ins.: cos 0 = 1

(d) u=21+3j+4k,v= -21-3j-4k Ans.:cos0= -1

(e) u = 21 + 3j, v = 31 + 4J Ins.: cos 8 =^18324

x/325=^325

5 Determine c so that the two given vectors will be orthogonal (or per-

pendicular).

(a) u=21-3j+4k,v=21+3j+ck In$.:c

(b)u=i+j+k, v=i+j+ck 4ns.:c=-2

6 For each vector v, find the unit vector v, in the direction of v.

(a) v=21-3j+4k Ins.:vl=3 i- 3 j+

4

k

(b) v = 71 .Ins.: vi = i

(c) v=1+j .Ins.:vs=_Li+ j

7 Supposing that v and w are orthonormal vectors and that u = av + bw,

where a and b are not both zero, find the angle between u and w. Hint: Use the

basic formulas

u.w - Jul Jw) cos 0, (av +

.Ins.: cos 0 - b/ a2 + b'.
8 With the text of this section out of sight, sketch a figure showing points
(0,0,0), Px(xs,yt,zi), and Ps(xs,ys,zs). Starting with the assumption that P(x,y,z)