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Properties of dot
products

2
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Properties of dot products
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2
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For any n-vectors v, w, w1 , and w2 , the following hold:
(i)
(ii)
(iii)
(iii0 )

v · w = w · v,
v · v = kvk2 ,
v · (cw) = c(v · w) for any scalar c, and v · (w1 + w2 ) = v · w1 + v · w2
...


To illustrate the type of work which
properties in Theorem 2
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1, assertion (i)
Pn explains the
Pgeneral
n
amounts to the calculation v · w =
i=1 vi wi =
i=1 wi vi = w · v
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5
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2
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1
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)
 
 
1
−2
Example 2
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2
...
Illustrating (i) above, we have
2
3
   
   
1
−2
−2
1
v · w = 3 ·  4  = 1 · (−2) + 3 · 4 + 2 · 3 = 16, w · v =  4  · 3 = (−2) · 1 + 4 · 3 + 3 · 2 = 16,
2
3
3
2
√
√
2
2
2
2
2
2
so v ·w = w ·v
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Finally, to illustrate (iii0 ) above, we have
    
    
      
1
−2
3
1
−10
6
1
−4
3 · 5  4  + 2 1 = 3 ·  20  + 2 = 3 ·  22  = −4 + 66 + 38 = 100
2
3
2
2
15
4
2
19
and

   
   
1
−2
1
3
5 3 ·  4  + 2 3 · 1 = 5 · 16 + 2 · 10 = 80 + 20 = 100
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2
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Let’s see that with the definition of angle we have given, for any nonzero n-vector x and
scalar c 6= 0, the angle between x and cx behaves as we would expect from experience in R2 and R3 :
• 0 radians (or 0°) if c > 0 (in this case x and cx “point in the same direction”);
• π radians (or 180°) if c < 0 (in this case x and cx “point in opposite directions”)
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2
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2
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Vectors making angles 0 and π with a nonzero vector x
The reason this works is just plugging into the definition, using Theorem 2
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1(ii),(iii), and recalling
(cx) · x
c(x · x)
c(x · x)
2
that cx has length |c|kxk
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Hence, if c > 0 then the angle
is arccos(1) = 0° and if c < 0 then it is arccos(−1) = 180° as desired
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That is, if x, y are
nonzero vectors in Rn for which the angle between them is either 0° or 180° (equivalently, x · y =
±kxkkyk) then we claim that necessarily y = cx for some scalar c (which is the same thing as x = by
for some scalar b: since x, y 6= 0 such a scalar multiplier in either case must be nonzero, so it can be
brought to the other side as its reciprocal)
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In Rn for general n,
this amounts to the second assertion in Theorem 2
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2 below (for those who are interested)
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2
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Visualization in R3 suggest that for any n, two unit vectors in
Rn or more generally two n-vectors v and w with the same length ` should be “close” when the
angle θ between them is near 0° and should be “unrelated” when the angle between them is
close to 90°
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Using the cosine of the angle as a measure of similarity for unit vectors in Rn for large n has
a striking real-world application: the natural language processing technique called latent
semantic analysis (also called latent semantic indexing)
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(Applying
this to a large sample of papers in materials science was used in [Tsh] to predict that specific
materials generate electricity from heat differences and so can improve fuel efficiency of cars,
power smart watches via body heat, etc
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(Most entries in each “word vector” are 0
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3) are applied to
the resulting huge collection of “word vectors”, yielding modified word vectors for which the
cosine of angles between them provides an effective way to measure similarity among parts of
documents
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Since the method is based entirely on mathematics and not on any linguistic information at
all (no dictionaries, no grammar, etc
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