SECTION
13.3
Dot Product and the Angle Between Two Vectors
(ET Section 12.3)
337
The cosine of the angle
α
between
e
θ
and the vector
i
in the direction of the positive
x
axis is
cos
=
e
·
i
k
e
k·k
i
k
=
e
·
i
=
((
cos
)
i
+
(
sin
)
j
)
·
i
=
cos
The solution of cos
=
cos
for angles between 0
◦
and 180
◦
is
=
. That is, the vector
e
makes an angle
with
the
x
axis. We now use the trigonometric identity
cos
cos
ψ
+
sin
sin
=
cos
(
−
)
to obtain the following equality:
e
·
e
=h
cos
,
sin
i·h
cos
,
sin
i=
cos
cos
+
sin
sin
=
cos
(
−
)
56.
Let
v
and
w
be vectors in the plane.
(a)
Use Theorem 2 to explain why the dot product
v
·
w
does not change if both
v
and
w
are rotated by the same angle
.
(b)
Sketch the vectors
e
1
= h
1
,
0
i
and
e
2
=
*
√
2
2
,
√
2
2
+
, and determine the vectors
e
0
1
,
e
0
2
obtained by rotating
e
1
,
e
2
through an angle
π
4
.Verifythat
e
1
·
e
2
=
e
0
1
·
e
0
2
.
SOLUTION
(a)
By Theorem 2,
v
·
w
=k
v
kk
w
k
cos
where
is the angle between
v
and
w
. Since rotation by an angle
does not change the angle between the vectors, nor
the norms of the vectors, the dot product
v
·
w
remains unchanged.
y
x
e
1
=
⟨
1, 0
⟩
e
2
=
⟨
,
⟩
2
2
2
2
(b)
Notice from the picture that if we rotate
e
1
by
/
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
0
1
=
D
√
2
2
,
√
2
2
E
and
e
0
2
= h
0
,
1
i
. Note that
e
1
·
e
2
=
1
·
√
2
2
+
0
·
√
2
2
=
√
2
2
and
e
0
1
·
e
0
2
=
0
·
√
2
2
+
1
·
√
2
2
=
√
2
2
. Thus,
e
1
·
e
2
=
e
0
1
·
e
0
2
.
57.
Determine
k
v
+
w
k
if
v
and
w
are unit vectors separated by an angle of 30
◦
.
We use the relation of the dot product with length and properties of the dot product to write
k
v
+
w
k
2
=
(
v
+
w
)
·
(
v
+
w
)
=
v
·
v
+
v
·
w
+
w
·
v
+
w
·
w
v
k
2
+
2
v
·
w
+k
w
k
2
=
1
2
+
2
v
·
w
+
1
2
=
2
+
2
v
·
w
We now Fnd
v
·
w
:
v
·
w
v
kk
w
k
cos 30
◦
=
1
·
1 cos 30
◦
=
√
3
2
Hence,
k
v
+
w
k
2
=
2
+
2
·
√
3
2
=
2
+
√
3
⇒k
v
+
w
k=
q
2
+
√
3
≈
1
.
93
58.
What is the angle between
v
and
w
if:
(a) v
·
w
=−k
v
kk
w
k
(b) v
·
w
=
1
2
k
v
w
k
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CHAPTER 13
VECTOR GEOMETRY
(ET CHAPTER 12)
(a)
Since
v
·
w
=k
v
kk
w
k
cos
θ
, it follows that
k
v
kk
w
k
cos
=−k
v
kk
v
k
cos
=−
1
The solution for 0
≤
≤
180
◦
is
=
180
◦
.
(b)
By
v
·
w
v
kk
w
k
cos
we get
k
v
kk
w
k
cos
=
1
2
k
v
kk
w
k
cos
=
1
2
The solution for 0
≤
≤
180
◦
is
=
60
◦
.
59.
Suppose that
k
v
k=
2and
k
w
3, and the angle between
v
and
w
is 120
◦
. Determine:
(a) v
·
w(
b
)
k
2
v
+
w
k
(c)
k
2
v
−
3
w
k
SOLUTION
(a)
We use the relation between the dot product and the angle between two vectors to write
v
·
w
v
kk
w
k
cos
=
2
·
3 cos 120
◦
=
6
·
µ
−
1
2
¶
3
(b)
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 Winter '08
 GANGliu
 Vectors, Dot Product, Cos

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