James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 12.5 Equations of Lines and Planes 823
53. Suppose that a ± 0.
(a) If a ? b − a ? c, does it follow that b − c?
(b) If a 3 b − a 3 c, does it follow that b − c?
(c) If a ? b − a ? c and a 3 b − a 3 c, does it follow
that b − c?
54. If v 1, v 2, and v 3 are noncoplanar vectors, let
k 1 −
v 2 3 v 3
v 1 ? sv 2 3 v 3d
k 2 −
v 3 3 v 1
v 1 ? sv 2 3 v 3d
(These vectors occur in the study of crystallography. Vectors
of the form n 1v 1 1 n 2v 2 1 n 3v 3 , where each n i is an integer,
form a lattice for a crystal. Vectors written similarly in terms of
k 1, k 2, and k 3 form the reciprocal lattice.)
(a) Show that k i is perpendicular to v j if i ± j.
(b) Show that k i ? v i − 1 for i − 1, 2, 3.
1
(c) Show that k 1 ? sk 2 3 k 3d −
v 1 ? sv 2 3 v .
3d
k 3 −
v 1 3 v 2
v 1 ? sv 2 3 v 3d
discovery Project
The geometry of a tetrahedron
P
A tetrahedron is a solid with four vertices, P, Q, R, and S, and four triangular faces, as shown in
the figure.
Q
S
R
1. Let v 1, v 2, v 3, and v 4 be vectors with lengths equal to the areas of the faces opposite the
vertices P, Q, R, and S, respectively, and directions perpendicular to the respective faces and
pointing outward. Show that
v 1 1 v 2 1 v 3 1 v 4 − 0
2. The volume V of a tetrahedron is one-third the distance from a vertex to the opposite face,
times the area of that face.
(a) Find a formula for the volume of a tetrahedron in terms of the coordinates of its vertices
P, Q, R, and S.
(b) Find the volume of the tetrahedron whose vertices are Ps1, 1, 1d, Qs1, 2, 3d, Rs1, 1, 2d,
and Ss3, 21, 2d.
3. Suppose the tetrahedron in the figure has a trirectangular vertex S. (This means that the three
angles at S are all right angles.) Let A, B, and C be the areas of the three faces that meet at S,
and let D be the area of the opposite face PQR. Using the result of Problem 1, or otherwise,
show that
D 2 − A 2 1 B 2 1 C 2
(This is a three-dimensional version of the Pythagorean Theorem.)
z
P¸(x¸, y¸, z¸)
a
P(x, y, z)
L
r¸ r
v
O
x
FIGURE 1
y
Lines
A line in the xy-plane is determined when a point on the line and the direction of the line
(its slope or angle of inclination) are given. The equation of the line can then be written
using the point-slope form.
Likewise, a line L in three-dimensional space is determined when we know a point
P 0 sx 0 , y 0 , z 0 d on L and the direction of L. In three dimensions the direction of a line is
con veniently described by a vector, so we let v be a vector parallel to L. Let Psx, y, zd be
an arbi trary point on L and let r 0 and r be the position vectors of P 0 and P (that is, they
have representations OP¸ A and OP l ). If a is the vector with representation P¸P A, as in Figure
1, then the Triangle Law for vector addition gives r − r 0 1 a. But, since a and v are
parallel vectors, there is a scalar t such that a − tv. Thus
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