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Unformatted text preview: the angle between a diagonal of a cube and one of its 2k c, c, c , ■ x, y, z , a
a 1, a 2 , a 3 , and b
that the vector equation r a
rb
sphere, and ﬁnd its center and radius. 1 3j 0 is Use this formula to ﬁnd the distance from the point
the line 3x 4 y 5 0. 3, 4, 5 30. c a x1 b y1 c
sa 2 b 2 3, 4 . 29. by one of its faces.
53. A molecule of methane, CH 4, is structured with the four hydro gen atoms at the vertices of a regular tetrahedron and the carbon atom at the centroid. The bond angle is the angle formed
by the H— C—H combination; it is the angle between the
lines that join the carbon atom to two of the hydrogen atoms.
Show that the bond angle is about 109.5 . [Hint: Take the
vertices of the tetrahedron to be the points 1, 0, 0 , 0, 1, 0 , 5E13(pp 848857) 850 ❙❙❙❙ 1/18/06 11:19 AM Page 850 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE 0, 0, 1 , and 1, 1, 1 as shown in the ﬁgure. Then the centroid
is ( 1 , 1 , 1 ).]
222 57. Use Theorem 3 to prove the CauchySchwarz Inequality: ab z a b H
58. The Triangle Inequality for vectors is
C H a a b (a) Give a geometric interpretation of the Triangle Inequality.
(b) Use the CauchySchwarz Inequality from Exercise 57 to
prove the Triangle Inequality. [Hint: Use the fact that
a b2
a b a b and use Property 3 of the
dot product.] y x b H H 54. If c ab
b a, where a, b, and c are all nonzero
vectors, show that c bisects the angle between a and b. 59. The Parallelogram Law states that 55. Prove Properties 2, 4, and 5 of the dot product (Theorem 2). a b 2 a b 2 2a 2 2b 2 56. Suppose that all sides of a quadrilateral are equal in length and (a) Give a geometric interpretation of the Parallelogram Law.
(b) Prove the Parallelogram Law. (See the hint in Exercise 58.) opposite sides are parallel. Use vector methods to show that the
diagonals are perpendicular.  13.4 The Cross Product
The cross product a b of two vectors a and b, unlike the dot product, is a vector. For
this reason it is also called the vector product. Note that a b is deﬁned only when a and
b are threedimensional vectors.
1 Definition If a
and b is the vector a a 1, a 2 , a 3 and b
b a 2 b3 b1, b2 ,...
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This note was uploaded on 02/04/2010 for the course M 56435 taught by Professor Hamrick during the Fall '09 term at University of Texas at Austin.
 Fall '09
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