What Are the Key Properties of Scalar Product and the Law of Cosines?

[Devil Moo]

Scalar Product is defined as $\mathbf A \cdot \mathbf B = | \vec A | | \vec B | \cos \theta$.

With the construct of a triangle, the Law of Cosines is proved.
$\mathbf A$ points to the tail of $\mathbf B$.
Well, $\mathbf C$ starts from the tail of $\mathbf A$ and points to somewhere.
Finally, $\mathbf B$ points to the head of $\mathbf C$.

Now someone defines a new operation such as $\mathbf A \circ \mathbf B = | \vec A | | \vec B | \sin \theta$. Following the same steps of proving the Law of Cosines.

$\begin{align} \mathbf C & = \mathbf A + \mathbf B \nonumber \\ \mathbf C \circ \mathbf C & = (\mathbf A + \mathbf B) \circ (\mathbf A + \mathbf B) \nonumber \\ | \vec C | | \vec C | \sin 0 & = | \vec A | | \vec A | \sin 0 + | \vec B | | \vec B | \sin 0 + 2 | \vec A | | \vec B | \sin \theta \nonumber \\ | \vec A | | \vec B | \sin \theta & = 0 \nonumber \end{align}$

$\theta = 0$, $\mathbf A = 0$ or $\mathbf B = 0$?

I am not quite understand what the result means. It seems the result is not consistent with the constructed triangle.

 

[fresh_42]

How is the angle defined? In the usual scalar product, this isn't important, because $\cos (\theta)=\cos (-\theta).$
This changes with the sine. I have the feeling that this could be a problem when adding $|A||B|\sin(\theta)$ and $|B||A|\sin(\theta)$ and pretend it would be commutative. Also the distributive property has to be proven.

 

[mfb]

Also the distribution law has to be proven.

Which is impossible, because the operation is not distributive.

The assumption of commutativity is the other mistake in post #1.

 

[Lucas SV]

This derivation is incorrect, because you assumed the operation you have defined $\circ$, is linear, in order to go from the second step to the third. Instead we have the identity.
$$
\mathbf A \times \mathbf B=(\mathbf A \circ \mathbf B) \mathbf n
$$
where $\mathbf n$ is a unit vector perpendicular to both $\mathbf A$ and $\mathbf B$, in the direction obtained from the right-hand rule (in the case $\mathbf A=\mathbf B$, you may pick any $\mathbf n$). Then, if $\mathbf C = \mathbf A + \mathbf B$,
$$
\mathbf C \times \mathbf C = (\mathbf A + \mathbf B) \times (\mathbf A + \mathbf B)
$$
Now the cross product $\times$ is linear, but you also get terms $\mathbf A \times \mathbf B$ and $\mathbf B \times \mathbf A$, which cancel by a property of the cross product. So in the end you get
$$
\mathbf C \times \mathbf C = \mathbf A \times \mathbf A + \mathbf B \times \mathbf B
$$
Which just says that $0=0+0$.

Another way to see that your conclusion does not hold, is simply to see that $\mathbf A \times \mathbf B$ is non-zero in general, so by taking the magnitude, we find that $|\mathbf A \circ \mathbf B|$ is non-zero in general, and hence $\mathbf A \circ \mathbf B$ is non-zero in general.

 

[Devil Moo]

How is the angle defined?

Now it is defined as the angle between $\mathbf A$ and $\mathbf B$ where $\theta$ is smaller than or equal to $\pi$ and it is commutative.

Which is impossible, because the operation is not distributive.

How do you find out that it is not distributive?

Another way to see that your conclusion does not hold, is simply to see that $\mathbf A \times \mathbf B$ is non-zero in general, so by taking the magnitude, we find that $|\mathbf A \circ \mathbf B|$ is non-zero in general, and hence $\mathbf A \circ \mathbf B$ is non-zero in general.

Um... How are these two examples related to the derivation?

 

[mfb]

How do you find out that it is not distributive?

By looking at an example.
$$A = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, B = \begin{pmatrix} -1 \\ 0 \\ 0 \end{pmatrix}, C = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$$
$$(A+B) \circ C = 0 \neq A \circ C + B \circ C$$

 

[fresh_42]

If you object, that you only need $A \circ (A+B) = A \circ B$ then compute it with $A=(1,2)\, , \, B=(1,3)$.

 

[mfb]

If you object, that you only need $A \circ (A+B) = A \circ B$ then compute it with $A=(1,2)\, , \, B=(1,3)$.

Huh?
That works, but this is not the only operation that was used in post 1.

 

[Devil Moo]

So if we have to define something, we have to calculate these laws before applying using the basic arithmetic rules.

 

[fresh_42]

You may define whatever you want.

1.) As long as it is well-defined.
This wasn't the case in your first attempt, because you assigned two possible values $|A||B| \sin(\theta)$ and $|A||B| \sin(-\theta)$ to $A \circ B$. A definition is not allowed to be ambiguous.

If this is given, it is absolutely legitimate to define, e.g. an operation $\circ$. But the next steps should then be to try and find out, which properties such an operation has. This includes algebraic rules it obeys. There might be new formulas, known ones that are partially true or even totally true, ones which would be nice to have but don't hold and so on.

2.) Any "basic rule" has to be proven before it is allowed to be applied.
This means that rules you may be used to by arithmetic from real numbers are not automatically true in the context of your new operation. Therefore they have to be proven first. E.g. matrix multiplication isn't commutative, squares of complex numbers aren't positive and so on.
You gave an example yourself: $A \circ A = 0$ although $A \neq 0$.

3) You might not need the entire basic rule.
E.g. you didn't need distribution in all cases for your argument. Only in some special cases. Therefore you didn't need to prove distribution, merely the ones you actually use. However, it doesn't hold even in these cases.

4.) You may summarize the above as follows: well-definition - toolbox - theorems.