Problern 3.4
Given A
=
*2
i3
+ 21 and B
=
tB' +y2 + 2Bz:
(a) find B' and B, if A is
Parallel
to B;
(b) find a rel4tion between
B' and B, If Lis perpendicular
to B'
Solution:
(a) If A is parallel
to B, then their directions are equal or opposite:
it
:
lin, or
From theycomPonent,
A/lAl
:
+B/lBl,
?293 +2
,
*,Bx+92+28,
___
frT


JT+4TE
.
l a
 )
 L
m:@
which can only be solved
for the minus sign (which means
that A and B must point
in opposite directions
for fhem to be parallel).Solving fot B! + Bl,
/
2
,'\
2
20
ni+ ni:
 \/r4

4:
;
.
\  J
/
9
8,
^
2\66
4
: :
I : 

v/i6D)
"r
3Jt4
3
From the )ccomponent,
2
J14
and,
from the zcomPonent,
,
Ar:A'
This
is consistent
with our result
for ni + A?
These
results
could also have been obtained
by assuming
04
was 0o or l80o and
solving
lAllBl
:
fA'B,
or by solving
A x B
=
0'
(rii if a
is perpendicular
to B, then
their dot product
is zero(see Section
31.4).
Using Eq.
(3.21),
0
:
A
'B :2Br 6+ Bz,
or
B,628,.
There are an
infinite number
of vectors which could be B and be perpendicular
to A,
but their x and zcomponents
must satisff
this
relation'
This result could have also been obtained
by assuming
0AB
:90"
and calculating
lAllBl
:lAxBl.
/ a \ r
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Problem
3.6
Given
vectors
A
:
i29*
23 andB
:
t3
tZ,finda
vector C whose
magnitude
is 9 and whose direction
is perpendicular
to both A and B.
Solution: The cross product of two vectors produces
a new vector which is
perpendicular
to both of the original vectors. Two vectors exist which have a
magnitude of 9 and are orthogonal
to both A and B: one which is 9 units l6ng in
the direction
of theunit vector
parallel
to A x B, and one
in theopposite
direction.
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 Spring '08
 RAKOV
 Cartesian Coordinate System, Electromagnet, Emoticon, ASCII art, Internet slang, LR

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