qazi (kaq87) – HW04 – berg – (55290)
1
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001
10.0points
Which of the following expressions are well
defined for all vectors
a
,
b
, and
c
?
I.
(
a
·
b
)
c
,
II.

a
 ·
(
b
·
c
) ,
III.
a
·
(
b
+
c
) .
1.
I only
2.
none of them
3.
II and III only
4.
II only
5.
all of them
6.
III only
7.
I and II only
8.
I and III only
correct
Explanation:
The dot product is defined only for two
vectors, and its value is a scalar.
On the
other hand, the length of any vector is always
a scalar.
For the three given expressions,
therefore, we see that
I.
is welldefined because it is the product
of a scalar and a vector.
II.
is not welldefined because the first term
in the dot product is not a vector.
III.
is welldefined because both terms in
the dot product are vectors.
keywords: vectors, dot product, T/F, length,
TrueFalse,
002
10.0points
Determine the dot product of the vectors
a
=
i
−
3
j
+ 2
k
,
b
= 2
i
−
j
−
2
k
.
1. a
·
b
= 5
2. a
·
b
= 3
3. a
·
b
= 1
correct
4. a
·
b
=
−
1
5. a
·
b
=
−
3
Explanation:
The dot product,
a
·
b
, of vectors
a
=
a
1
i
+
a
2
j
+
a
3
k
,
b
=
b
1
i
+
b
2
j
+
b
3
k
is defined by
a
·
b
=
a
1
b
1
+
a
2
b
2
+
a
3
b
3
.
Consequently, when
a
=
i
−
3
j
+ 2
k
,
b
= 2
i
−
j
−
2
k
,
we see that
a
·
b
= 1
.
003
10.0points
Which of the following statements are true for
all vectors
a
,
b
?
A.

a
·
b
 ≤ 
a
 
b

,
B.

a
+
b

2
=

a

2
+ 2
a
·
b
+

b

2
,
C.

a
·
b

=

a

b

=
⇒
a
⊥
b
.
1.
B and C only
2.
A and B only
correct
3.
A and C only
4.
B only
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qazi (kaq87) – HW04 – berg – (55290)
2
5.
all of them
6.
none of them
7.
A only
8.
C only
Explanation:
If
θ
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 Spring '07
 Gilbert
 Linear Algebra, Multivariable Calculus, Vectors, Vector Space, Dot Product, Euclidean vector, qazi

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