While I was reading a book on mechanics, when introducing the vector multiplication the author stated that multiplying two vectors can produce a vector, a scalar, or some other quantity.
1.4 Multiplying Vectors
Multiplying one vector by another could produce a vector, a scalar, or some other quantity. The choice is up to us. It turns out that two types of vector multiplication are useful in physics.
An Introduction to Mechanics, Daniel Kleppner and Robert Kolenkow
The authors then examine the scalar or "dot product" and the vector or "cross product" (the latter not shown in the above link; can be seen on Amazon's preview) but seem to make no mention of any other method.
My concern is not about vector multiplication here, but what can be that quantity which is neither a scalar nor a vector. The author has explicitly remarked the quantity as neither a scalar nor a vector.
What I think is that, when we define a vectors and scalars, we propose the definition in terms of direction. In one direction is considered and in another it is not considered. Then, how can this definition leave space for any other quantity being as none of the two?
I would be obliged if someone could explain me if the statement is correct and how is it so. Also it would be great if you can substantiate your argument using examples.
Best Answer
If you have two vectors $\mathbf{a}$ and $\mathbf{b}$, the inner product $\mathbf{a} \cdot \mathbf{b}$ is a scalar, the cross product $\mathbf{a} \times \mathbf{b}$ is a vector and the dyadic product $\mathbf{a} \otimes \mathbf{b}$ is a matrix. It is defined as
$$\mathbf{a}\otimes\mathbf{b} = \mathbf{a b}^\mathrm{T} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}\begin{pmatrix} b_1 & b_2 & b_3 \end{pmatrix} = \begin{pmatrix} a_1b_1 & a_1b_2 & a_1b_3 \\ a_2b_1 & a_2b_2 & a_2b_3 \\ a_3b_1 & a_3b_2 & a_3b_3 \end{pmatrix} $$
It occurs a lot in the formalism of quantum mechanics where it is written as $|a \rangle \langle b|$ (using the so-called bra-ket notation by Dirac).
With regard to direction: if you apply a matrix to a vector, the vector may get stretched / compressed along multiple axes. So in contrast to a vector, a matrix involves multiple directions.