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Why can't we create new vector multiplications like:

$$\vec{A}\odot\vec{B}=|A||B|\sin \theta$$

and

$$\vec{A}\otimes\vec{B}=AB\cos \theta \, \hat{p}$$

where $\hat{p}$ is a unit vector perpendicular to vectors $\vec{A}$ and $\vec{B}$.

Why don't we get any physical quantities from such vector multiplication?

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    But $\sin\theta$ is always a scalar, no? – 2017-02-27
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    What do you mean? – 2017-02-27
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    That $\sin\theta$ is a scalar, not a vector, despite what you say. Which word do you fail to understand in my first comment? – 2017-02-27
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    If you flip the roles of $\sin \theta$ and $\cos \theta$, the nature [**inner product space**](https://en.wikipedia.org/wiki/Inner_product_space) will loose. For example, conjugate symmetry and positive-definiteness. – 2017-02-27
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    Because magnitude times magnitude times sine of something (a scalar) is again a scalar (as "magnitudes" are scalars...) , just as if we multiply magnitude times magnitude times **cosine** of something we again get a scalar – 2017-02-27
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    Well this isn't exactly true. We can replace $cos(x)$ with $sin(x +\pi/2)$ if you were so inclined and it would give the same result. – 2017-02-27

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You can - you can always define new stuff in math (if they're defined properly), but you need to ask yourself is it interesting. Can you make a whole new inner product theory with your 'Scalar Product' (with other implications and theorems)? Maybe. Probably, some people tried to explore that area and didn't find something worth publishing or sharing with the world, or they did and it didn't get a lot of attention.