How to change the order of the three vectors that are dot product

Question

Two vectors: $u$ (a unit vector) and $x$ (a regular non-unit vector), the angle between them is $\theta$

I want to know the derivation process of changing $(x\cdot u)u$ to ***x (intuitively, I want to move $x$ to the far right side, and the ***is any proper form that makes it work).

One of the appealing reasons to do that is that I can use *** as an operation on vector x.

But here, does anyone could help me out for how to derive it from $(x\cdot u)u$u to ***x ?

Note that $u(u\cdot x)$ is not what we want, there should be no parenthesis.

Answer 1

If we define $ (x \cdot u) = x^T u $, then $ (x \cdot u) u = (x^T u) u = k u$. However, note that $ k = x^T u $ is a scalar. Equivalently, $ k = u^T x $.

$$ (x \cdot u) u = k u $$.

Since $k$ is an scalar we can exchange the positions of $k$ and $u$. Then, $$ (x \cdot u) u = u k = u (u^T x) $$

Finally

$$ (x \cdot u) u = P x $$ where $ P = u u^T $ is a matrix.

Answer 2

This is impossible. $(x \cdot u)u$ is a scalar multiple of $u$, and you can't write it as a multiple of $x$ unless $x$ is a multiple of $u$.

For instance, let $u=i$ and $x=i+j$. Then $(x \cdot u)u = i$, and you can't write $i$ as a multiple of $i+j$.

Also, scalar multiplication by a vector is not a one-to-one operation, so “canceling” a dot product or multiplying by an inverse to a vector is undefined.

Answer 3

Using Einstein notation: \begin{align} ((x \cdot u)u)_i &=x_ju_ju_i\\ &=u_iu_jx_j\\ &=P_{ij}x_j\\ &=(Px)_i \end{align} So we have that $$(x \cdot u)u = Px$$ If we define the $P$ matrix as $P_{ij}=u_iu_j$. Note that because $u$ is an unit vector, $P$ will be the projection matrix to $u$.