Calculus: Analytic Geometry and Calculus, with Vectors

2.3 Scalar products, direction cosines, and lines in E3 73

lying in the same plane, it is sometimes convenient to think of this plane as being
the plane of the paper upon which we print or write and to use the right-handed
coordinate system shown in Figure 2.392. Vectors of the form v = xi + yj + zk
for which z = 0 then lie in the plane of the paper and figures showing them are

Y

/.i

k i

J Y

Figure 2.391 Figure 2.392

y

i

Figure 2.393

undistorted. When we are interested only in vectors lying in one plane, we may

leave the z axis out of the figure and use Figure 2.393. The introduction is

finished, and we come to our problem. Use the method, involving slopes, of

Section 1.3 to show that tan 0 = when 0 is the angle between the vectors

u=2i+3j, v=3i+4j

running from the origin to the points (2,3) and (3,4). Use the method of this

section to show (or to show again) that

18 = 324

cos0=

325 325

Then construct and use a modest but appropriate figure to show that the two

results agree. To conclude with another story, we can remark that the method

involving slopes may sometimes be preferred because it often gives answers

without radicals when easy problems in E2 are solved. However, the scalar-

product method is the more powerful method which works in E2 and in Ea and,

as some people learn, in E when it > 3.

13 A vector v makes equal acute angles 6 with the three positive coordinate

axes. Find 3 (to find cos 3 is enough) (i) by use of an identity involving direction

cosines and (ii) by using the edges and the diagonals of a cube having opposite

vertices at (0,0,0) and (1,1,1). Make everything check.

14 Referring to Figure 2.21, find the angle 0 between the two vectors running

from the origin to the mid-points of the top edges of the cube that pass through

P(3,3,3). Ans.: cos 0 = $.

15 Let v be a nonzero vector and w a unit vector having their tails at the

same point 0. Show with the aid of a figure that the vector (vw)w is the vector

component of v in the direction of w, and that the vector v - (vw)w is the

vector component of v orthogonal (or perpendicular) to w. Remark: This prob-

lem appeared among Problems 2.19 with different notation and additional

information.

16 When u 0 0, each vector v is representable as the sum of a vector com-

ponent cu and a vector component q orthogonal to u. Find q and find a way