ET 12.3; M 13.3. Dot Product
Arithmetic definition.
In
R
3
, the
scalar product
of vectors
~a
and
~
b
is
~a
·
~
b
=
a
1
b
1
+
a
2
b
2
+
a
3
b
3
.
Connections with geometry.
For lengths,
~a
·
~a
=

~a

2
.
On directions, two nonzero vectors are perpen
dicular if and only if their dot product is equal to 0.
When
~a
6
=
~
0, the
vector projection
proj
~
a
~
b of
~
b
onto
~a
has the form
s~a
, where
s
=
(
~
b
·
~a
)
(
~a
·
~a
)
=
~
b
·
~a

~a

2
.
Also,
s~a
=

~
b

(cos
θ
)ˆ
a
=

~
b

(cos
θ
)
~a

~a

,
where
θ
is the smallest nonegative angle between the
vectors
~
b
and
~a
. It follows that
~
b
·
~a
=

~
b

~a

cos
θ,
and that
cos
θ
=
~
b
·
~a

~
b

~a

.
1
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ET 12.4; M 13.4. Cross Product
Special fact in 3 dimensions.
If
~a
and
~
b
are
not
aligned, then every vector that is perpendicular to
both
~a
and
~
b
is a multiple of the vector
(
a
2
b
3

a
3
b
2
)
ˆ
i
+ (
a
3
b
1

a
1
b
3
)
ˆ
j
+ (
a
1
b
2

a
2
b
1
)
ˆ
k.
This
vector product
~a
×
~
b
can also be computed in
other ways.
Magnitude and direction.
The
length
of
~a
×
~
b
is

~a

~
b

sin
θ,
for the same angle
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 Spring '03
 Unknown
 Vectors, Scalar, Vector Space, Dot Product, nonzero vectors, θ

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