Unformatted text preview: orce is applied in the direction 0, 3,
at the end of the wrench. Find the magnitude of the force
needed to supply 100 J of torque to the bolt. v2
v1 4 5 j and let u be a vector with length 3 that starts at
the origin and rotates in the xy-plane. Find the maximum and v1
v1 v3 k2 v3 v1 v1
v2 v3 v2
v2 v3 (These vectors occur in the study of crystallography. Vectors of
the form n1 v1 n 2 v2 n3 v3 , where each ni is an integer, form
a lattice for a crystal. Vectors written similarly in terms of k1,
k2, and k3 form the reciprocal lattice.)
(a) Show that k i is perpendicular to vj if i j.
(b) Show that k i vi 1 for i 1, 2, 3.
(c) Show that k1 k2 k3
v1 v2 v3 5E-13(pp 858-867) 858 ❙❙❙❙ 1/18/06 11:26 AM Page 858 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE DISCOVERY PROJECT
The Geometry of a Tetrahedron
A tetrahedron is a solid with four vertices, P, Q, R, and S, and four triangular faces as shown in
the ﬁgure. P 1. Let v1 , v2 , v3 , and v4 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
v1 v2 v3 v4 0 S
R Q 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 P 1, 1, 1 , Q 1, 2, 3 , R 1, 1, 2 ,
and S 3, 1, 2 .
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,
D 2 A2 B 2 C 2
(This is a three-dimensional version of the Pythagorean Theorem.) |||| 13.5 Equations of Lines and Planes z P¸(x¸, y¸, z¸)
P(x, y, z) L r¸
O r v x
y FIGURE 1 A line in the xy-plane is determined when a point on the l...
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