Tensor product of 3 vectors I understand how to do a tensor product of 2 vectors, but I am not quite sure how to incorporate a third one.
Can someone explain to me how to compute tensor product if I have 3 vectors
$$u\otimes v\otimes w$$
$$u=(1,1)$$
$$v=(1,-2)$$
$$w=(-1,3)$$
I can't find any examples online, and I feel like I will understand it most if I see an example.
Thank you.
 A: Well, it's easier if you write them in index notation
$u^{i} = \begin{bmatrix}1\\1\end{bmatrix} $
$v^{i} = \begin{bmatrix}1\\-2\end{bmatrix} $
$w^{i} = \begin{bmatrix}-1\\3\end{bmatrix} $
then their tensor product $(u\otimes v\otimes w)$ simply has components:
$$(u\otimes v\otimes w)^{ijk} = u^{i}v^{j}w^{k}$$
To be completely explicit:
$$\begin{array}
(u\otimes v\otimes w)^{111} = u^{1}v^{1}w^{1} &= (1)(1)(-1) &= -1\\
(u\otimes v\otimes w)^{112} = u^{1}v^{1}w^{2} &= (1)(1)(3) &= 3\\
(u\otimes v\otimes w)^{121} = u^{1}v^{2}w^{1} &= (1)(-2)(-1) &= 2\\
(u\otimes v\otimes w)^{122} = u^{1}v^{2}w^{2} &= (1)(-2)(3) &= -6\\
(u\otimes v\otimes w)^{211} = u^{2}v^{1}w^{1} &= (1)(1)(-1) &= -1\\
(u\otimes v\otimes w)^{212} = u^{2}v^{1}w^{2} &= (1)(1)(3) &= 3\\
(u\otimes v\otimes w)^{221} = u^{2}v^{2}w^{1} &= (1)(-2)(-1) &= 2\\
(u\otimes v\otimes w)^{222} = u^{2}v^{2}w^{2} &= (1)(-2)(3) &= -6\\
\end{array}$$
So you see, in order to write the components of this tensor as a matrix you would need to arrange them in a cube. But the components are there.
