How exactly can I convert the below equation into the vector form? (i.e. V(i,j,k) form or $a*i+b*j+c*k$ form): $$\frac{x-5}{-10}=\frac{y-3}{-6}=\frac{z-2}{-4}$$
I'm actually trying to find the angle between two 3D lines, but I only know how to find out angles between vectors, so I'm trying to convert the above equation to vector form so as to carry out what I need to.
You should realise first that vectors are always drawn from the origin, whereas lines do not have to go through the origin. If you understand this, then what you're asking for is pretty simple, set a parameter $t$ equal to the line equation you have, so:
$$ {x-5\over -10} = {y-3\over -6} = {z-2\over -4} = t $$
Then you just make three simultaneous equations:
$$ x = -10t + 5, \qquad y = -6t +3 , \qquad z = -4t +2 $$
Then you can write this in vector form directly as:
$$ \begin{bmatrix} x \\ y \\ z \end{bmatrix} = t \begin{bmatrix} -10 \\ -6 \\ -4 \end{bmatrix} + \begin{bmatrix} 5 \\ 3 \\ 2 \end{bmatrix}$$
Edit
Note however, that you can not find the angle between two lines, unless they are touching, because by definition, they do not intersect. What you can do however, is find the angle between their "direction vectors". In this case, the direction vector is:
$$\begin{bmatrix} -10 \\ -6 \\ -4 \end{bmatrix}$$
This is the analog of the gradient of a line in 2D, because it tells you the direction that the line moves in.