# Become a Linear Algebra Expert

subsection{}

Scalar equation of a plane in $mathbb{R}^3$ is described by the general equation,

begin{equation}

n_1x_1 + n_2x_2 +n_3x_3 = d

end{equation}

where $(n_1,n_2,n_3)$ are the coordinates of the vector that is normal to the plane.

This equation can be derived if we know the normal vector, and a known point on the plane, $P(p_1,p_2,p_3)$. Then, we define any general point on plane, $Q(x_1,x_2,x_3)$.

First, we create $vec{PQ}$.

begin{equation}

vec{PQ} = begin{bmatrix}x_1-p_1x_2-p_2x_3-p_3end{bmatrix}

end{equation}

We know that $vec{PQ}$ is on the plane, and will be orthogonal to the normal vector. Which means,

$vec{n} cdot vec{PQ} = 0$

Expanding it out,

begin{equation}

begin{split}

n_1(x_1-p_1)+n_2(x_2-p_2)+n_3(x_3-p_3)=0

n_1x_1 + n_2x_2 +n_3x_3 – (n_1p_1+n_2p_2+n_3p_3)= 0

end{split}

end{equation}

Since $(n_1p_1+n_2p_2+n_3p_3)$ are given numbers, it is possible to group them as another real number, $d$.

begin{equation}

begin{split}

n_1x_1 + n_2x_2 +n_3x_3 – d = 0

n_1x_1 + n_2x_2 +n_3x_3 = d

end{split}

end{equation}

where $d = ncdot p$.

Expanding this same concept to $mathbb{R}^n$,

begin{equation}

n_1x_1 + n_2x_2 +n_3x_3…n_nx_n = d

end{equation}

In this case, it is not called scalar equation of a plane, but scalar equation of a hyperplane.