Understanding the multiplication of vectors (Part 1)
Approach by applying the notion of multiplication of real numbers to
vectors to prove that the real numbers multiplication cannot be applied to
vectors
⃑ and ⃑𝒃 be two different vectors.
Let 𝒂
Since we know that additions of vectors are defined and one is able to
calculate the quantity of such calculation, we can derive from it that a vector
can be multiplied by a number and produces a quantity. For instance, since
𝑎 + 𝑎 is defined, one can have the following relationship such that 𝑎 + 𝑎 =
2 ∗ 𝑎, where vector 𝑎 is multiplied by a number and such calculation is
defined. In such instances, the result quantity of it is a vector that has
double the magnitude and the same direction of 𝑎. Thus, if a vector is
multiplied by a number, that is, if a quantity of magnitude and direction is
multiplied by a quantity of magnitude, the quantity of magnitude can only
affect the magnitude of the quantity of magnitude and direction and has no
affect on the direction of the quantity of magnitude and direction. Also,
since only the magnitude of the quantity of magnitude and direction is
affected while its direction is not affected, and the result change in its
magnitude is shown as unaffected by other factors such directions, one can
conclude that the direction of the quantity of magnitude and direction has
no affect on the magnitude of the other quantity.
With the similar idea, one could predict the result quantity of the calculation
of multiplying two quantity of magnitude and direction. Let 𝑎 and 𝑏⃑ be the
two quantities of magnitude and direction. According to the conclusion in
the discussion above, since quantities 𝑎 and 𝑏⃑ have both magnitude and
direction, the magnitude of 𝑎 should affect and only affect the magnitude of
𝑏⃑, and similarly, the direction of 𝑎 should affect and only affect the direction
of 𝑏⃑. That is, to multiply the two vectors 𝑎 and 𝑏⃑, one should multiply both
their magnitude and direction individually. In order to do calculations on
such quantities, we need to express such quantities numerically; in this
instance, we need to express the magnitude and the direction of the two
quantities separately. Let |𝑎| be the magnitude of vector 𝑎, |𝑏⃑| be the
magnitude of vector 𝑏⃑. Since both vectors are quantities with both
magnitude and direction, the direction of them can be shown as such
quantities without magnitude; that is, a quantity with a non-existing
magnitude and an existing direction. Since the magnitude of such quantity
does not exist, in order to do calculations about it for one, one needs a
numerical expression of such non-existence according to the calculation it is
in. In additions, non-existence is equivalent to zero; in multiplication, nonexistence is equivalent to one, since the existence of these such numbers
does not have effects on the original calculations. However, such model
produces two essential problems:
1. A quantity with direction but without magnitude carries no
significance, since any calculation including addition and multiplication
done by such quantity to another quantity has no affect at all. Since
such quantity does not have magnitude, it cannot affect other
quantity’s magnitude and direction. It cannot change other’s
magnitude because a non-existing magnitude ought to have no affect
on other’s magnitude through any calculation; it cannot change
other’s direction because a change in direction requires calculations
done by a quantity with both magnitude and direction: according to
the geometric definition of additions of vectors, since one vector has
no magnitude, it cannot change the position of the initial vector’s
terminal point, thus the result vector remains the same as the initial
vector. Since multiplication is essentially the calculation of repeated
addition, such conclusion can be applied to multiplication as well. Thus
a quantity with direction and without magnitude cannot change other
vectors’ direction.
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2. The multiplication of direction and direction is not defined. According
to the discussion above, in order to multiply one vector to another,
one should multiply their magnitudes and directions individually since
the multiplication between magnitude and direction is not defined.
For the multiplication of magnitudes, since magnitudes are real
numbers, it can apply the same rule of multiplication of real number
and thus it is well defined. Yet for the multiplication of directions, such
calculation cannot be defined. The proof of it can be stated as the
following:
Let O be a vector that has a magnitude of zero. By the definition of
multiplication between vectors and real numbers, when a vector is
multiplied by a real number n, its magnitude is multiplied by n with its
direction unchanged. Thus, any vector multiplied by zero produces the
vector O; that is, 0 ∗ 𝑛⃑ = 𝑂. Since the range of choice of the vector 𝑛⃑
that is multiplied is any vector, the direction of O can be any direction
and thus it is in indeterminate form. Similar to infinity where one
infinity can be larger or smaller than the other, one O can have
different direction than the other; since the direction of such vector O
is indeterminate, its direction is undefined, and thus no calculations
can be applied to its direction.
Therefore, because of the problems above appeared in the multiplication of
vectors as we apply the notion of multiplication of real numbers to vectors,
the multiplication of vectors cannot be defined; in other words, the
multiplication applied for real number failed to be applied to vectors. Since
so, the special new multiplication for vectors must be introduced in order to
compute the multiplication of vectors.
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