The vectors a and b are direction vectors. The intersection point is a, well, position vector. Direction vectors are just directions; they don't have a beginning, end, or any point in-between. You can calculate the length of a direction vector, and you can calculate the angle between 2 direction vectors (at least in 2D), but you cannot calculate their intersection point just because there is no concept like a position when looking at direction vectors.

On the other hand, a ray can be defined as

r( t ) := p + t * d

where p denotes any point on the ray (obviously a position vector), d denotes the direction of the ray (obviously a direction vector), and t denotes the running parameter.

Accordingly to one of the rules dealing with positions and directions, namely

position + direction => position

a ray denotes an infinitesimal amount of points (if d is not null).

So you have 2 rays, each one with its own running parameter

r1( t1 ) := p1 + t1 * d1

The above is a linear equation system with 2 equations in 2 variables t1 and t2, and hence can be solved the usual way:

p1 + t1 * d1 = p2 + t2 * d2

<=>

p1 - p2 = t2 * d2 - t1 * d1

Using an abbreviation for the difference between the both origins of the rays

c := p1 - p2

cx = t2 * d2x - t1 * d1x

cy = t2 * d2y - t1 * d1y

Solving this for e.g. t1 gives

cx * d2y - cy * d2x = t1 * ( d1y * d2x - d1x * d2y )

Comparing this with the working of the perpDot operator, one can see the correspondences, so that the above is the same as

perpDot( c, d2 ) = t1 * perpDot( d1, d2 )

I still need to get the value of either t1 or t2 to go any further, don't I?

Yes. But notice that either t1 or t2 is identical to the t in your OP.

Solving a linear equation system means to look at all equations, and to use all other equations to eliminate all other unknowns until just 1 unknown is left. In your problem you have 2 equations in the system when you write down the vector equation as scalar equation. You also have 2 unknowns, namely t1 and t2. Assuming that the both equations are not linear dependent, they are sufficient to solve the problem.

Looking in particular at the equation system

cx = t2 * d2x - t1 * d1x

cy = t2 * d2y - t1 * d1y

Multiply e.g. the upper equation with d2y and the lower equation with d2x yields in

cx * d2y = t2 * d2x * d2y - t1 * d1x * d2y

cy * d2x = t2 * d2y * d2x - t1 * d1y * d2x

Do you need to compute the 2nd unknown t2 as well? No, because of r1( t1 ) = r2( t2 ) means that you'd get the same point a second time.

The vectors a and b are direction vectors. The intersection point is a, well, position vector. Direction vectors are just directions; they don't have a beginning, end, or any point in-between. You can calculate the length of a direction vector, and you can calculate the angle between 2 direction vectors (at least in 2D), but you cannot calculate their intersection point just because there is no concept like a position when looking at direction vectors.

Oh, boy. Sorry, the mathematician in me needs to make this paragraph more precise, even if nobody else cares.

There are vectors (what I think haegarr is calling "direction vectors") and there are points (what I think he's calling "position vectors"). Vectors don't have a beginning, end or any point in between. As long as you have a metric defined (if you don't know what this means, it doesn't matter, because you are probably thinking of Euclidean space, which comes with a metric), you can calculate the length of a vector and you can calculate the angle between 2 vectors (in any dimension), but you cannot calculate their intersection point.

But nothing in this problem really requires a metric, because it's a perfectly well defined affine problem, so we shouldn't have to talk about length and angle at all.

What the OP wants is to compute the intersection between two rays in 2D. A ray is specified by a starting point S and a vector d, and it is the set of all points of the form P = S + t*d, with t >= 0.

The standard name for that operation is determinant of a 2x2 matrix, whose columns are the two vectors. The solution of a system of linear equations as the ratio of two determinants is known as Cramer's rule.

Sometimes I'm lazy and just skip to the answer rather than work it out myself. Here's a very good site about all sorts of geometry-related calculations:

http://paulbourke.net/geometry/

Add it to your bookmarks :)