homogeneous coordinates I am to use homogeneous coordinates to calculate a standard matrix for a projection onto the line $4x-2y=6$ from the point $(3,10)$.
I'm not sure what homogeneous coordinates are and neither how to use them to calculate my problem? Please believe me, I've been looking for information on various sources!

In 2D points in homogeneous coordinates have the form $P = (x,y,1)$ and lines $L=(a,b,c)$ such that the equation for the line can be found by
$$ L \cdot P = 0 \} a x + b y + c = 0 $$
So in your case, the homogeneous coordinates for the point is $P=(3,10,1)$ and the line $L=(4,-2,-6)$.
The trick with homogeneous coordinates is that if you multiply them with a scalar value, it does not change the underlying geometry. This is powerful be because the coordinates of a point $P=(a,b,c)$ are $(x,y) = (\frac{a}{c},\frac{b}{c})$. To test if a point belongs to a line simply do the dot product $P\cdot L=0$ and check for zero. The coordinates of a line joining two points $P$, $Q$ are
$$\begin{pmatrix} p_x\\p_y\\1 \end{pmatrix} \times \begin{pmatrix} q_x\\q_y\\1 \end{pmatrix} = \begin{pmatrix} p_y-q_y \\ q_x-p_x \\ p_x q_y - p_y q_x \end{pmatrix} $$
and similarly the point where two lines $L=(a,b,c)$ and $M=(u,v,w)$ meet is
$$\begin{pmatrix} a\\b\\c \end{pmatrix} \times \begin{pmatrix} u\\v\\w \end{pmatrix} = \begin{pmatrix} b w - c v \\ c u-a w \\ a v - b u \end{pmatrix} $$
The above has coordinates $(x,y) = \left( \frac{b w -c v}{a v - b u} , \frac{c u-a w}{a v - b u} \right)$ which if you do the calculations with vectors you will end up with the same value.
Futhermore, the minimum distance of the line $L=(a,b,c)$ to the origin is $d = -\frac{c}{\sqrt{a^2+b^2}}$ and the point closest to the origin has coordinates $P=(-a c,-b c, a^2+b^2) \equiv \left(-\frac{a c}{a^2+b^2},-\frac{b c}{a^2+b^2}\right)$
In addition, the minimum distance of the line $L=(a,b,c)$ to the point $P=(u,v,w)$ is
$$ {\rm dist}(L,P) = \frac{a u + b v + c w}{w \sqrt{a^2+b^2}} $$
and the projected point on the line has coordinates
$$ {\rm proj}(L,P) = \left( b( b u-a v)-a c, -a ( b u - a v)- b c, w (a^2+b^2) \right) $$
An affine 3×3 transfromation can be defined for translations and rotations.
$$  \begin{bmatrix} 1 & 0 & t_x \\ 0 & 1 & t_y \\ 0 & 0 & 1 \end{bmatrix} \begin{pmatrix} p_x\\p_y\\1 \end{pmatrix} =  \begin{pmatrix} p_x+t_x\\p_y+t_y\\1 \end{pmatrix}$$
$$  \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{pmatrix} p_x\\p_y\\1 \end{pmatrix} =  \begin{pmatrix} p_x \cos\theta-p_y \sin \theta\\p_x \sin\theta + p_y \cos\theta\\1 \end{pmatrix}$$
With these basics you can built what you need from here.
