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34. The tension T at each end of the chain has magnitude 25 N. What is the weight of the chain? 37° light ray given by the vector a
a 1, a 2 , a 3 ﬁrst strikes the
xz-plane, as shown in the ﬁgure. Use the fact that the angle of
incidence equals the angle of reﬂection to show that the direction of the reﬂected ray is given by b
a 1, a 2 , a 3 . Deduce
that, after being reﬂected by all three mutually perpendicular
mirrors, the resulting ray is parallel to the initial ray. (American
space scientists used this principle, together with laser beams
and an array of corner mirrors on the Moon, to calculate very
precisely the distance from the Earth to the Moon.)
z 37° 35. If A, B, and C are the vertices of a triangle, ﬁnd l
CA. b 36. Let C be the point on the line segment AB that is twice as far from B as it is from A. If a
that c 2 a 1 b.
OA, b l
OB, and c l
OC, show a
x y 5E-13(pp 838-847) 1/18/06 11:15 AM Page 843 SECTION 13.3 THE DOT PRODUCT |||| 13.3 ❙❙❙❙ 843 The Dot Product
So far we have added two vectors and multiplied a vector by a scalar. The question arises:
Is it possible to multiply two vectors so that their product is a useful quantity? One such
product is the dot product, whose deﬁnition follows. Another is the cross product, which
is discussed in the next section.
a 1, a 2 , a 3 and b
1 Definition If a
and b is the number a b given by
ab b1, b2 , b3 , then the dot product of a a 1 b1 a 2 b2 a 3 b3 Thus, to ﬁnd the dot product of a and b we multiply corresponding components and
add. The result is not a vector. It is a real number, that is, a scalar. For this reason, the dot
product is sometimes called the scalar product (or inner product). Although Deﬁnition 1
is given for three-dimensional vectors, the dot product of two-dimensional vectors is
deﬁned in a similar fashion:
a 1, a 2 b1, b2 a 1 b1 a 2 b2 EXAMPLE 1 2, 4 3,
6, 2, i 2j 1
2 2j 1, 7, 4 1 k 3k 23 4 1 16
22 3 ) 6 1 7 1...
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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