338
C H A P T E R
13
VECTOR GEOMETRY
(ET CHAPTER 12)
y
x
e
1
=
⟨
1, 0
⟩
e
2
=
⟨
,
⟩
2
2
2
2
(b)
Notice from the picture that if we rotate
e
1
by
π
/
4, we get
e
2
, and when we rotate
e
2
by the same amount we get
a unit vector along the
y
axis. Thus,
e
1
=
√
2
2
,
√
2
2
and
e
2
=
0
,
1 . Note that
e
1
·
e
2
=
1
·
√
2
2
+
0
·
√
2
2
=
√
2
2
and
e
1
·
e
2
=
0
·
√
2
2
+
1
·
√
2
2
=
√
2
2
. Thus,
e
1
·
e
2
=
e
1
·
e
2
.
57.
Determine
v
+
w
if
v
and
w
are unit vectors separated by an angle of 30
◦
.
SOLUTION
We use the relation of the dot product with length and properties of the dot product to write
v
+
w
2
=
(
v
+
w
)
·
(
v
+
w
)
=
v
·
v
+
v
·
w
+
w
·
v
+
w
·
w
=
v
2
+
2
v
·
w
+
w
2
=
1
2
+
2
v
·
w
+
1
2
=
2
+
2
v
·
w
We now find
v
·
w
:
v
·
w
=
v
w
cos 30
◦
=
1
·
1 cos 30
◦
=
√
3
2
Hence,
v
+
w
2
=
2
+
2
·
√
3
2
=
2
+
√
3
⇒
v
+
w
=
2
+
√
3
≈
1
.
93
58.
What is the angle between
v
and
w
if:
(a) v
·
w
= −
v
w
(b) v
·
w
=
1
2
v
w
SOLUTION
(a)
Since
v
·
w
=
v
w
cos
θ
, it follows that
v
w
cos
θ
= −
v
v
cos
θ
= −
1
The solution for 0
≤
θ
≤
180
◦
is
θ
=
180
◦
.
(b)
By
v
·
w
=
v
w
cos
θ
we get
v
w
cos
θ
=
1
2
v
w
cos
θ
=
1
2
The solution for 0
≤
θ
≤
180
◦
is
θ
=
60
◦
.
59.
Suppose that
v
=
2 and
w
=
3, and the angle between
v
and
w
is 120
◦
. Determine:
(a) v
·
w
(b)
2
v
+
w
(c)
2
v
−
3
w
SOLUTION
(a)
We use the relation between the dot product and the angle between two vectors to write
v
·
w
=
v
w
cos
θ
=
2
·
3 cos 120
◦
=
6
·
−
1
2
= −
3
(b)
By the relation of the dot product with length and by properties of the dot product we have
2
v
+
w
2
=
(
2
v
+
w
)
·
(
2
v
+
w
)
=
4
v
·
v
+
2
v
·
w
+
2
w
·
v
+
w
·
w
=
4
v
2
+
4
v
·
w
+
w
2
We now substitute
v
·
w
= −
3 from part (a) and the given information, obtaining
2
v
+
w
2
=
4
·
2
2
+
4
(
−
3
)
+
3
2
=
13
⇒
2
v
+
w
=
√
13
≈
3
.
61

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S E C T I O N
13.3
Dot Product and the Angle Between Two Vectors
(ET Section 12.3)
339
(c)
We express the length in terms of a dot product and use properties of the dot product. This gives
2
v
−
3
w
2
=
(
2
v
−
3
w
)
·
(
2
v
−
3
w
)
=
4
v
·
v
−
6
v
·
w
−
6
w
·
v
+
9
w
·
w
=
4
v
2

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