Pythagoras On Dot and Cross Products
The dot and cross products are often introduced via trigonometric
functions and/or matrix operations, but they also arise quite
naturally from consideration of Pythagoras' theorem. Given two
points v and V in the 3D vector space with Cartesian coordinates
(x,y,z) and (X,Y,Z) respectively, the squared distance between
these two points is
S^2 = (x-X)^2 + (y-Y)^2 + (z-Z)^2
Also, if (and only if) these two vectors are perpendicular, the
distance between them is the hypotenuse of a right triangle with
edge lenghts equal to the lengths of the two vectors. Thus we
have
[S^2]_perp = (x^2 + y^2 + z^2) + (X^2 + Y^2 + Z^2)
if and only if v and V are perpendicular. Therefore, the necessary
and sufficient condition for v and V to be perpendicular is if S^2
equals [S^2]_perp. Equating these two expressions and cancelling
terms, we arrive at the necessary and sufficient condition
xX + yY + zZ = 0
This motivates the definition of the left hand quantity as the "dot
product" (also called the scalar product) of the vectors (x,y,z) and
(X,Y,Z).
At the other extreme, suppose we want to determine whether the
vectors v and V are parallel. In any case the squared length of
the vector SUM of the two vectors is obviously
S^2 = (x+X)^2 + (y+Y)^2 + (z+Z)^2
If (and only if) the two vectors are parallel we have |S| = |v| + |V|,
which implies
[S^2]_para = v^2 + 2|v||V| + V^2
= (x^2 + y^2 + z^2) + (X^2 + Y^2 + Z^2) + 2|v||V|
Hence the necessary and sufficient condition for v and V to be
parallel is for S^2 to equal [S^2]_para. Equating these two
expressions, cancelling terms, and squaring both sides gives
(xX + yY + zZ)^2 = (x^2 + y^2 + z^2)(X^2 + Y^2 + Z^2)
Expanding these expressions and cancelling terms, we have the
condition
2xXyY + 2xXzZ + 2yYzZ = (xY)^2 + (xZ)^2 + (yX)^2
+ (yZ)^2 + (zX)^2 + (zY)^2
Notice that we can gather terms and re-write this equality as
(xY - Xy)^2 + (Xz - xZ)^2 + (yZ - Yz)^2 = 0
This sum of squares equals zero only if each term is individually
zero, which of course was to be expected, because two vectors are
parallel if and only if their components are in the same proportions
to each other, i.e.,
x/y = X/Y x/z = X/Z y/z = Y/Z
which represents the vanishing of the three terms in the previous
expression. This motivates the definition of the cross product (also
known as the vector product) of two vectors (x,y,z) and (X,Y,Z) as
consisting of those three components, ordered symmetrically, so that
each component is defined in terms of the other two components of the
arguments, i.e.,
[(yZ - Yz), (Xz - xZ), (xY - Xy)]
This vector is null if and only if v and V are parallel. Furthermore,
notice that the dot product of this cross product and the vector v
is identically zero, i.e.,
xyZ - xYz + yXz - yxZ + zxY - zXy = 0
and likewise the dot product of the cross product and V is also
identically zero
XyZ - XYz + YXz - YxZ + ZxY - ZXy = 0
As we saw previously, the dot product of two vectors is 0 if and only
if the vectors are perpendicular, so this shows that the cross product
of v and V is perpendicular to both of them.
Generalizations to Higher Dimensions
It's also interesting to consider what happens in more than three
dimensions. Clearly the same argument as above applies to the dot
product in four or more dimensions, yielding a scalar quantity that
equals zero if and only if the two n-dimensional vectors are
orthogonal. However, the cross product is less striaghtforward.
If we repeat the above argument in, say, four dimensions, to find
the condition for two vectors
v = (w,x,y,z) V = (W,X,Y,Z)
to be parallel, the result is
(wX - Wx)^2 + (wY - Wy)^2 + (wZ - Wz)^2
+ (xY - Xy)^2 + (Xz - xZ)^2 + (yZ - Yz)^2 = 0 (1)
Again this was to be expected, since the vanishing of these terms
corresponds to the proportionality of the respective components of
v and V. For example, the first term represents w/x = W/X. However,
we now have SIX such conditions, so we can't use these as the
components of a four-dimensional vector. In general, when we
form this kind of "cross-product" of two n-dimensional vectors we
arrive at C(n,2) terms. It so happens that C(3,2) equals 3, so the
cross product of two three-dimensional vectors is a three-dimensional
vector (albiet with a sign ambiguity). On the other hand, C(4,2)
equals 6, so the "cross product" of two four-dimensional vectors
(defined in this way) is a SIX dimensional vector. It would do no
good to simply expand our domain to six dimensions, because the
"cross product" of two 6D vectors would have C(6,2) = 15 components.
Of course, we don't necessarily require all six of the components
of the cross product of two four-dimensional vectors, since only
three of them can serve to force proportionality on all four
components of the original vectors. However, we can't arrive at
a symmetrical choice of three (or four) terms, so this approach
doesn't assure us that the cross product is orthogonal to the
two given vectors. Furthermore, it's clear that the orthogonality
requirement is under-specified by two vectors in four dimensions,
because there is an entire plane of directions, each orthogonal
to any two given vectors.
This might lead us to define the cross product in four dimensions
as an operation on THREE vectors, instead of just two. On this
basis we can specify that the cross product has the unique direction
(up to sign) required to make it perpendicular to each of the
three given vectors. In addition, we could define the magnitude
of the cross-product as the volume contained inside the parallel-
epiped whose edges (at one vertex) are the three given vectors.
This directly generalizes the fact that the magnitude of the
cross product of two vectors in three dimensions equals the area
of the parallelogram induced by the two given vectors. Clearly
we can generalize this to any number of dimensions, with the
understanding that this "cross product" operates on n-1 vectors.
On the other hand, suppose we're intent on defining an operation
on two vectors in four-dimensional space, and we want it to have
the properties that (1) the result is a four-dimensional vector,
(2) the result is perpendicular to both of the given vectors, and
(3) the magnitude of the result vanishes if the two given vectors
are parallel. This can be accomplished, but not in a unique way,
by combining, for each component of the "cross product", the three
terms from equation (1) that don't involve that component, with
appropriate choices of signs. For example, we can define an ersatz
cross-product of v and V as
[ (xY-Xy)+(xZ-Xz)+(yZ-Yz), (Yz-yZ)+(yW-Yw)+(zW-Zw),
(xZ-Xz)+(Xw-xW)+(zW-Zw), (Xy-xY)+(Xw-xW)+(Yw-yW) ]
We can choose the signs of the twelve "ab-ba" expressions in any one
of eight distinct ways (up to overall sign), giving eight equally
suitable ersatz cross products in 4D space. The eight sets of
signs that satisfy the requirements are, letting "1" denote positive
and "0" denote negative,
111 111 111 111 (shown above)
011 101 101 111
111 011 011 011
011 001 001 011
101 110 111 101
001 100 101 101
101 010 011 001
001 000 001 001
So, for any two given 4D vectors v and V, the ersatz cross product
can be any one of these eight (up to sign), all of which lie on
a single 2D plane (perpendicular to both v and V). For example,
with v=(19,-29,27,-13) and V=(-5,-11,16,23) the results are shown
below
It's worth noting that although each of these eight cross products
vanishes when v and V are parallel, the converse is not true. In
general the condition for true parallelism can be expressed as
w/y = W/Y
x/y = X/Y
z/y = Z/Y
which forces the components to all be in the same proportion to
each other. However, the necessary and sufficient condition for
the ersatz cross product to vanish is that there must exist a
constant k such that
w/y = W/Y + k(1 + W/Y)
x/y = X/Y + k(1 + X/Y)
z/y = Z/Y + k(1 + Z/Y)
With k=0 this gives true parallelism, but we can also have other
values of k, such as 1/2, which explains why the ersatz cross product
of v=(5,2,1,8) and V=(3,1,1,5) is (0,0,0,0). Interestingly, the
condition for pseudo-parallelism implies the matrix equation
_ _ _ _ _ _
| || | | |
| 0 (Y+Z) (Z-X) -(X+Y) || w | | 0 |
| || | | |
| -(Y+Z) 0 (W-Z) (Y+W) || x | | 0 |
| || | = | |
| -(Z-X) -(W-Z) 0 -(X-W) || y | | 0 |
| || | | |
| (X+Y) -(Y+W) (X-W) 0 || z | | 0 |
|_ _||_ _| |_ _|
which resembles the anti-symmetric form of the Lorentz-invariant
electromagnetic tensor.
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