13
VECTOR GEOMETRY
13.1 Vectors in the Plane
(ET Section 12.1)
Preliminary Questions
1.
Answer true or false. Every nonzero vector is:
(a)
Equivalent to a vector based at the origin.
(b)
Equivalent to a unit vector based at the origin.
(c)
Parallel to a vector based at the origin.
(d)
Parallel to a unit vector based at the origin.
SOLUTION
(a)
This statement is true. Translating the vector so that it is based on the origin, we get an equivalent vector based at the
origin.
(b)
Equivalent vectors have equal lengths, hence vectors that are not unit vectors, are not equivalent to a unit vector.
(c)
This statement is true. A vector based at the origin such that the line through this vector is parallel to the line through
the given vector, is parallel to the given vector.
(d)
Since parallel vectors do not necessarily have equal lengths, the statement is true by the same reasoning as in (c).
2.
What is the length of
−
3
a
if
k
a
k=
5?
Using properties of the length we get
k−
3
a
k=−
3
k
a
3
k
a
3
·
5
=
15
3.
Suppose that
v
has components
h
3
,
1
i
. How, if at all, do the components change if you translate
v
horizontally two
units to the left?
Translating
v
=h
3
,
1
i
yields an equivalent vector, hence the components are not changed.
4.
What are the components of the zero vector based at
P
= h
3
,
5
i
?
The components of the zero vector are always
h
0
,
0
i
, no matter where it is based.
5.
Are the following true or false?
(a)
The vectors
v
and
−
2
v
are parallel.
(b)
The vectors
v
and
−
2
v
point in the same direction.
(a)
The lines through
v
and
−
2
v
are parallel, therefore these vectors are parallel.
(b)
The vector
−
2
v
is a scalar multiple of
v
, where the scalar is negative. Therefore
−
2
v
points in the opposite direction
as
v
.
6.
Explain the commutativity of vector addition in terms of the Parallelogram Law.
To determine the vector
v
+
w
, we translate
w
to the equivalent vector
w
0
whose tail coincides with the
head of
v
. The vector
v
+
w
is the vector pointing from the tail of
v
to the head of
w
0
.
vv
'
w
'
w
v
+
w
To determine the vector
w
+
v
, we translate
v
to the equivalent vector
v
0
whose tail coincides with the head of
w
.Then
w
+
v
is the vector pointing from the tail of
w
to the head of
v
0
. In either case, the resulting vector is the vector with the
tail at the basepoint of
v
and
w
, and head at the opposite vertex of the parallelogram. Therefore
v
+
w
=
w
+
v
.
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View Full DocumentSECTION
13.1
Vectors in the Plane
(ET Section 12.1)
157
Exercises
1.
Sketch the vectors
v
1
,
v
2
,
v
3
,
v
4
with tail
P
and head
Q
, and compute their lengths. Are any two of these vectors
equivalent?
v
1
v
2
v
3
v
4
P
(
2
,
4
)
(
−
1
,
3
)
(
−
1
,
3
)
(
4
,
1
)
Q
(
4
,
4
)
(
1
,
3
)
(
2
,
4
)
(
6
,
3
)
SOLUTION
Using the deFnitions we obtain the following answers:
v
1
=
−→
PQ
=h
4
−
2
,
4
−
4
i=h
2
,
0
i
k
v
1
k=
p
2
2
+
0
2
=
2
y
x
Q
P
v
1
v
2
1
−
(
−
1
),
3
−
3
2
,
0
i
k
v
2
p
2
2
+
0
2
=
2
y
x
Q
P
v
2
v
3
2
−
(
−
1
),
4
−
3
3
,
1
i
k
v
3
p
3
2
+
1
2
=
√
10
y
x
Q
P
v
3
v
4
6
−
4
,
3
−
1
2
,
2
i
k
v
4
p
2
2
+
2
2
=
√
8
=
2
√
2
y
x
Q
P
v
4
v
1
and
v
2
are parallel and have the same length, hence they are equivalent.
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 Fall '09
 Grabovsky
 Calculus, Linear Algebra, Vectors, Vector Space, Dot Product, E C T O R G E O M E T

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