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Vectors  Part 2:
Products
Dot Product
(scalar product)
If the angle between two vectors
A
r
and
B
r
is
θ
, then we define the dot product
cos( )
AB AB
θ
⋅=
r
r
This product is a scalar.
The angle between two vectors is always between 0
and 180°.
This product obeys the commutative and distributive rules.
Special cases:
θ
= 0
(parallel vectors):
A
BA
B
r
r
(the product of the magnitudes, positive)
θ
= 90° (perpendicular):
0
AB
r
r
θ
= 180° (antiparallel):
A
B
−
r
r
(negative)
2
A
AA
rr
(any vector, dotted with itself, = the magnitude squared)
Applied to unit vectors:
ˆˆ
ˆˆ ˆˆ
1
ii jj kk
⋅=⋅=⋅=
ˆˆ ˆ
ˆ
0
ij jk k
i
This makes it easy to find the dot product of vectors in component form.
For
example,
()
(
)
ˆ
ˆ
23
5
2
5
3
1
7
iji
j
−⋅+
=
⋅
−
⋅
=
One use of the dot product is to find the angle between two vectors.
From the
definition above, the angle
θ
between two vectors is found from
cos( )
AB
⋅
=
r
r
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View Full DocumentAs an example, let’s find the angle between the face diagonal and body diagonal
of a cube.
Let the cube have side = s with the edges aligned in a Cartesian CS.
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 Spring '10
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