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Point direction form:

$$ \frac{xx_1}{a} = \frac{y  y_1}{b} = \frac{zz_1}{c} $$ 
Two point form:

$$ \frac{xx_1}{x_2x_1} = \frac{y  y_1}{y_2y_1} = \frac{zz_1}{z_2z_1} $$ 
Parametric form:

$$ \begin{aligned} x &= x_1 +t\,\cos \alpha \\ y &= y_1 +t\,\cos \beta \\ z &= z_1 +t\,\cos \gamma \end{aligned} $$ 
The distance from $P_2(x_2,y_2,z_2)$ to the line through $P_1(x_1,y_1,z_1)$ in the direction $(a,b,c)$ is

$$ d = \sqrt{ \frac{\left[c(y_2y_1)b(z_2z_1)\right]^2 + \left[a(z_2z_1)c(x_2x_1)\right]^2 + \left[b(x_2x_1)a(y_2y_1)\right]^2} {a^2 + b^2 + c^2 } } $$ 
The distance between two lines. First one through $P_1(x_1,y_1,z_1)$ in direction $(a_1,b_1,c_1)$, Second one: through $P_2(x_2,y_2,z_2)$ in direction $(a_2,b_2,c_2)$ is:

$$ d = \frac{ \begin{vmatrix} x_2x_1 & y_2y_1 & z_2z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} } { \sqrt{\begin{vmatrix} b_1 & c_1 \\ b_2 & c_2 \end{vmatrix}^2 + \begin{vmatrix} c_1 & a_1 \\ c_2 & a_2 \end{vmatrix}^2 + \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}^2 }} $$ 
The two lines intersect if:

$$ \begin{vmatrix} x_2x_1 & y_2  y_1 & z_2  z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} = 0$$ 
Please tell me how can I make this better.