Least squares problem: find the line through the origin in $\mathbb{R}^{3}$

The problem is as follows:

“Please set up (but do not solve) the normal equations for the following least squares
approximation problem: Find $(a, b, c, d)$ such that the plane H described by $ax + by + cz = d$
minimizes $\sum |ax_i + by_i + cz_i − d|^2$ where the $(x1, y1, z1), \cdots ,(x6, y6, z6)$ are the following points: $(2,3,4)$, $(99,−85,0)$, $(0,1,8)$, $(5,2,2)$, $(3,3,3)$, $(1,2,4)$.”

I solved a similar problem with points representing ordered pairs, for example $(2,1)$, $(3,5)$ and so on, but the question was to get a line that approximates such ordered pairs. In that case I represented the points as $y = mx + b$ and got $m$ and $b$ for the new line. I wonder if for the new problem which is a 3D space I also have to represent the ordered triplets as $z = ax + by + c$ or something like that.

I will very much appreciate any clue.

Solutions Collecting From Web of "Least squares problem: find the line through the origin in $\mathbb{R}^{3}$"

You are on the right track. Set up your equations like you would in 2 dimensions.
\begin{align*}
z_1 &= d + ax_1 + by_1 \\
z_2 &= d + ax_2 + by_2 \\
&\vdots \\
z_6 &= d + ax_6 + by_6.
\end{align*}

From this, you can hopefully derive the normal equations as matrix products.

We have our data points $\left\{ x_{k}, y_{k}, z_{k}, d_{k} \right\}_{k=1}^{m}$, and our trial function, a line in $\mathbb{R}^{3}$ through the origin:
$$
d\left( x, y, z \right) = a x + b y + c z.
$$

The linear system is
$$
\begin{align}
\mathbf{A} a &= d \\
%
\left[ \begin{array}{ccc}
x & y & z
\end{array} \right]
%
\left[ \begin{array}{c}
a
\end{array} \right]
%
& =
%
\left[ \begin{array}{c}
d
\end{array} \right]\\[3pt]
%
\left[ \begin{array}{ccc}
x_{1} & y_{1} & z_{1} \\
x_{2} & y_{2} & z_{2} \\
\vdots & \vdots & \vdots \\
x_{m} & y_{m} & z_{m} \\
\end{array} \right]
%
\left[ \begin{array}{c}
a \\ b \\ c
\end{array} \right]
%
&=
%
\left[ \begin{array}{c}
d_{1} \\ d_{2} \\ \vdots \\ d_{m}
\end{array} \right].
%
\end{align}
$$

The normal equations are
$$
\begin{align}
\mathbf{A}^{*} \mathbf{A} a &= \mathbf{A}^{*} d \\
\left[
\begin{array}{ccc}
x \cdot x & x \cdot y & z \cdot z \\
y \cdot x & y \cdot y & y \cdot z \\
x \cdot z & x \cdot z & z \cdot z \\
\end{array}
\right]
%
\left[
\begin{array}{c}
a \\
b \\
c
\end{array}
\right]
%
&=
%
\left[
\begin{array}{c}
x \cdot d \\
y \cdot d \\
z \cdot d
\end{array}
\right].
%
\end{align}
%
$$
The solution is
$$
a_{LS} =
%
\left[
\begin{array}{c}
a \\
b \\
c
\end{array}
\right]_{LS}
%
= \left( \mathbf{A}^{*}\mathbf{A} \right)^{-1} \mathbf{A}^{*} d.
$$