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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.

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I will very much appreciate any clue.

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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.

$$

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