Math 23
Sections 110113
B. Dodson
Week 2 Homework:
12.2 vectors: unit, standard unit, notations
12.3 dot product: orthogonal, proj, comp
12.4 cross product: formula, properties
Problem 12.2.25:
Find a unit vector
±u
that has the same direction
as
±a
= 8
±
i

±
j
+ 4
±
k.
[variation/continuation: ﬁnd a vector of length 4
in the opposite direction.]
Solution:
The length

8
±
i

±
j
+ 4
±
k

=
√
64 + 1 + 16 =
√
81 = 9
,
so the unit vector is
±u
=
1

±a

±a
=
1
9
‡
8
±
i

±
j
+ 4
±
k
·
=
8
9
±
i

1
9
±
j
+
4
9
±
k.
Likewise, the vector with length 4 is

4
9
‡
8
±
i

±
j
+ 4
±
k
·
.
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Problem 12.3.23bc:
Determine whether the given vectors are
othogonal, parallel or neither.
(b)
±a
=
<
4
,
6
>,
±
b
=
<

3
,
2
> .
(c)
±a
=

±
i
+ 2
±
j
+ 5
±
k,
±
b
= 3
±
i
+ 4
±
j

±
k.
Solution:
(b) Recall that vectors are
parallel
if
one is a scalar multiple of the other,
±a
=
c
±
b,
for a scalar
c.
This says
<
4
,
6
>
=
c <

3
,
2
>
=
<

3
c,
2
c >,
so 4 =

3
c,
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 Spring '06
 YUKICH
 Vectors, Dot Product, 2J, 4k, 1 j, 5k, 4j

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