58 Vectors and geometry in three dimensions
22 Using the definition of Problem 21, prove that if(1) Clvl + C2v2 + C3v3 = 0,
where ci, C2, c3 are scalars that are not all zero, then some one of the vectors
v1j v2, v3 belongs to the span of the other two. Remark: In this case the set of
three vectors vi, v2, v8 is said to be a dependent (or linearly dependent) set. In
case (1) holds only when cl = c2 = C3 = 0, the three vectors are said to be inde-
pendent. These concepts are very important in several branches of mathematics.
23 Perhaps we need a little experience drawing and adding vectors that all
lie in the same plane. Start with a clean sheet of paper and draw unit vectors
u and v headed, respectively, toward the right side and top of the page. Let
PO be the point at the center of the page. More points in the sequence Po,
P1f P2, P3, are to be obtained in the following random way. Start with
k = 1. Get two coins of different size and toss them so that each lands H (head)
or T (tail).
If big coin is H and small coin is H, let Pk_1Pk = U.
If big coin is H and small coin is T, let Pk_1Pk = v.
If big coin is T and small coin is H, let Pk_1Pk = -U.
If big coin is T and small coin is T, let Pk_1Pk = -V.
Then draw PoPI. With k = 2, repeat the coin tossing to locate P2, and continue
until P1o has been reached. It is not improper to become interested in the proba-
bility that all of the points Po, P1, .. , P10lie inside the circle with center at
the origin and radius 5. This is a random-walk problem and such problems are
of interest in the theory of diffusion. To prepare for investigation of these
things, we must study analytic geometry, calculus, probability, and statistics.
24 Using one die (singular of dice, a cube with six numbered faces) instead
of two coins, describe a procedure for obtaining paths for use in random-walk
problems in E3.
25 The problem here is to grasp the meanings of the following statements
when n is 2 and 3 and perhaps even when n is a greater integer. When P1, P2,
, P.+1 are n + 1 points that lie in the same E. but do not lie in an E.1,
these points are the vertices of an n-dimensional simplex. A line segment which
joins two of these points is an edge of the simplex, so the simplex has n(n + 1)/2edges. To each vertex Pk there corresponds the opposite simplex of n - 1
dimensions having vertices at the remaining points. A median ofa simplex is
the line segment joining a vertex Pk to the centroid 4 of the opposite simplex.
The n + 1 medians of the simplex all intersect at a point B, and for each k,PbB =n+1 Pk1k.
This point B is the centroid of the n-dimensional simplex and, whenan origin 0
has been selected, the centroid B is determined by the formulaOB =OPl + OP2 + OP3 +... +
n + 1
Remark: As the assertions may have suggested, simplexes ofone, two, and three
dimensions are, respectively, line segments, triangles, and tetrahedrons. When