Thomas Calculus 13th [Solutions]

888 Chapter 12 Vectors and the Geometry of Space
20 . Suppose the diagonals of a rectangle are perpendicular, and let u and v be the sides of a rectangle
the diagonals are d1
u v and d2 u v . Since the diagonals are perpendicular we have d1 d2 0
2 2
( u v) ( u v) u u u v v u v v 0 v u 0 v u v u 0 v u 0
which is not possible, or v u 0 which is equivalent to v u the rectangle is a square.
21. Clearly the diagonals of a rectangle are equal in length. What is not as obvious is the statement that equal
diagonals happen only in a rectangle. We show this is true by letting the adjacent sides of a parallelogram be
the vectors v1i v2j and u1i u2 j . The equal diagonals of the parallelogram are
d1 v1i v2j u1i u2j and d2 v1i v2 j u1i u2 j . Hence d1 d2 v1i v2j u1i u2j
2 2
v1i v2 j u1i u2j v1 u1 i v2 u2 j v1 u1 i v2 u2 j v1 u1 v2 u2
2 2 2 2 2 2 2 2 2 2
v1 u1 v2 u2 v1 2v1u 1 u1 v2 2v2u2 u2 v1 2v1u 1 u1 v2 2v2u2 u2
2 v1u 1 v2u2 2 v1u 1 v2u2 v1u 1 v2u2 0 v1i v2j u1i u2j 0 the vectors v1i
v2j
and u1i
u2j are perpendicular and the parallelogram must be a rectangle.
22. If u v and u v is the indicated diagonal, then
2 2
( u v) u u u v u u v u u v | v|
u v v v ( u v)
v the angle
cos
1
( u v)
u
u v u
between the diagonal and u and the angle
1 ( u v)
v
cos
between the diagonal and v are equal because the inverse cosine function is one-to-one.
u v v
Therefore, the diagonal bisects the angle between u and v.
23. horizontal component: 1200 cos(8 ) 1188 ft/s; vertical component: 1200 sin(8 ) 167 ft/s
24.
2.5 lb
w cos 33 15 2.5 lb, so w . Then
cos18
w
2.5 lb
cos18
cos33 , sin 33 2.205, 1.432
25. (a) Since cos 1, we have u v u v cos u v (1) u v .
(b) We have equality precisely when cos 1 or when one or both of u and v is 0. In the case of nonzero
vectors, we have equality when 0 or , i.e. , when the vectors are parallel.
26. xi yj v xi yj v cos 0 when 2
.
This means ( x, y ) has to be a point whose position
vector makes an angle with v that is a right angle or
bigger.
27.
2 2
v u1 au1 bu2 u1 au1 u1 bu2 u1 a u1 b u2 u1
a(1) b(0)
a
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