$$
\begin{align}
\color{#C00000}{((10,10,6)-(5,5,4))}\times\color{#00A000}{((10,10,3)-(5,5,5))}
&=\color{#C00000}{(5,5,2)}\times\color{#00A000}{(5,5,-2)}\\
&=\color{#0000FF}{(-20,20,0)}
\end{align}
$$
is perpendicular to both lines; therefore, $\color{#0000FF}{(-20,20,0)}\cdot u$ is constant along each line. If this constant is not the same for each line, the lines do not intersect. In this case, the constant for each line is $0$, so the lines intersect.
$$
\color{#C00000}{(5,5,2)}\times\color{#0000FF}{(-20,20,0)}=\color{#E06800}{(-40,-40,200)}
$$
is perpendicular to the first line; therefore, $\color{#E06800}{(-40,-40,200)}\cdot u$ is constant along the first line. In this case, that constant is $400$. The general point along the second line is
$$
\color{#00A000}{(5,5,5)}+\color{#00A000}{(5,5,-2)}t
$$
To compute the point of intersection, find the $t$ so that
$$
\color{#E06800}{(-40,-40,200)}\cdot(\color{#00A000}{(5,5,5)}+\color{#00A000}{(5,5,-2)}t)=400\\
600-800t=400\\
t=1/4
$$
Plugging $t=1/4$ into the formula for a point along the second line, yields the point of intersection:
$$
\color{#00A000}{(5,5,5)}+\color{#00A000}{(5,5,-2)}\cdot1/4=\left(\frac{25}{4},\frac{25}{4},\frac{9}{2}\right)
$$
The second example
$$
\begin{align}
\color{#C00000}{((12,15,4)-(6,8,4))}\times\color{#00A000}{((12,15,6)-(6,8,2))}
&=\color{#C00000}{(6,7,0)}\times\color{#00A000}{(6,7,4)}\\
&=\color{#0000FF}{(28,-24,0)}
\end{align}
$$
is perpendicular to both lines; therefore, $\color{#0000FF}{(28,-24,0)}\cdot u$ is constant along each line. If this constant is not the same for each line, the lines do not intersect. In this case, the constant for each line is $-24$, so the lines intersect.
$$
\color{#C00000}{(6,7,0)}\times\color{#0000FF}{(28,-24,0)}=\color{#E06800}{(0,0,-340)}
$$
is perpendicular to the first line; therefore, $\color{#E06800}{(0,0,-340)}\cdot u$ is constant along the first line. In this case, that constant is $-1360$. The general point along the second line is
$$
\color{#00A000}{(6,8,2)}+\color{#00A000}{(6,7,4)}t
$$
To compute the point of intersection, find the $t$ so that
$$
\color{#E06800}{(0,0,-340)}\cdot(\color{#00A000}{(6,8,2)}+\color{#00A000}{(6,7,4)}t)=-1360\\
-680-1360t=-1360\\
t=1/2
$$
Plugging $t=1/2$ into the formula for a point along the second line, yields the point of intersection:
$$
\color{#00A000}{(6,8,2)}+\color{#00A000}{(6,7,4)}\cdot1/2=\left(9,\frac{23}{2},4\right)
$$