Least squares fitting of vectors I want to find:
$argmin_{\lambda} = \sum_{i\epsilon I}\left \| \vec{P_{i}} - \lambda\vec{Q_{i}} \right \|^2$
where $P = (x^{'}, y^{'}, z^{'})$ and $Q = (x^{''}, y^{''}, z^{''})$ are  representations of 3-dimensional displacements.
I get the general concept of least squares fitting, and I can solve the problem $Af=B$ in the least square sense by doing $A^{t}A\hat{f}=A^{t}B$, but that is just with 2 dimensions.
How would it work if $\vec{P}$ and $\vec{Q}$ are tridimensional vectors?
 A: The method of least squares works for any dimension (assuming invertibility of $A^TA$).
$$
Ax = b\\
A^TA\hat{x} = A^Tb\\
\hat{x} = (A^TA)^{-1}A^Tb
$$
For your problem you might do the following:
$$
    Q_x = \begin{bmatrix}Q_{1x} & Q_{2x} & \dots &Q_{nx}\end{bmatrix}_{1\times n}\\
    Q_y = \begin{bmatrix}Q_{1y} & Q_{2y} & \dots &Q_{ny}\end{bmatrix}_{1\times n}\\
    Q_z = \begin{bmatrix}Q_{1z} & Q_{2z} & \dots &Q_{nz}\end{bmatrix}_{1\times n}\\
    P_x = \begin{bmatrix}P_{1x} & P_{2x} & \dots &P_{nx}\end{bmatrix}_{1\times n}\\
    P_y = \begin{bmatrix}P_{1y} & P_{2y} & \dots &P_{ny}\end{bmatrix}_{1\times n}\\
    P_z = \begin{bmatrix}P_{1z} & P_{2z} & \dots &P_{nz}\end{bmatrix}_{1\times n}\\
    q = \begin{bmatrix}Q_x & Q_y & Q_z\end{bmatrix}^T\ _{3n\times1} \\
    p = \begin{bmatrix}P_x & P_y & P_z\end{bmatrix}^T\ _{3n\times1}
$$
With such matrices it's easy to solve the problem. We want:
$$
   q\lambda = p
$$
Such $\lambda$ probably doesn't exist. Using least squares we get $\lambda$ that minimizes the square error:
$$
    \hat{\lambda} = (q^Tq)^{-1}q^Tp = \frac{q\circ p}{q\circ q}
$$
