Nah, s'okay. I kinda rushed that explanation and it wasn't the best I've done. I don't have my textbooks any more because I had to give them back, but I'll try my best to remember how it all works. This post will be updated as I write more of it.
A vector is a quantity with both magnitude (size) and direction. They are sometimes written as columns of numbers like vertical co-ordinates, but since I cant do that easily on here, I'll write co-ordinates in standard text and vectors in bold text. Therefore, (1,2,3) is a co-ordinate, (1,2,3) is a vector for the purposes of this explanation.
A point has a position vector which is equal to its co-ordinates. So, the point (2,6,4) has position vector (2,6,4) relative to (0,0,0). That is to say, to get from (0,0,0) to (2,6,4) one must travel 2 units in the X direction, 6 units in the Y direction and 4 units in the Z direction. More generally, the point (a,b,c) has position vector (a,b,c) and is reached by travelling a units in X, b units in Y and c units in Z.
To find the vector that moves FROM the point (1,2,3) TO the point (4,5,6), one must calculate (4,5,6)-(1,2,3). This is simple: treat each part of the vector separately to the rest. So, (4,5,6)-(1,2,3) = (4-1, 5-2, 6-3) = (3,3,3). To move in the other direction, calculate (1,2,3)-(4,5,6) =(-3, -3, -3).
In general, the vector PQ that moves FROM P(a,b,c) TO Q(d,e,f) is (d-a, e-b, f-c). The vector that moves FROM Q to P is equal to -PQ.
Now consider the line going through points with position vectors (a, b, c) and (d, e, f). Obviously P(a, b, c) and Q(d, e, f) are on that line. So, a general point R(x, y, z) on that line will have a position vector equal to the position vector of either P or Q plus some other quantity.
Now imagine the line in 3D space: To reach any point on the line, you first move to a point on it, then either toward or away from another point on it. Think of it like a ring on a bar - first you put the ring on the bar, then you can slide it along the bar, but not away from the bar. A convenient movement along the line is a multiple of the vector that goes from P to Q. Therefore, the point R has position vector OP + mPQ. Make sense, or do you want me to explain better?
More later.
