To get to what a cross product truly is:
In Physics, we use this to determine Torque and Angular Momentum. For Torque, take the cross product of the Force Vector and the Radius of the disc, or vector from the point of application to the center. (t = F x R)
The Torque vector will lie in a plane perpendicular to both the Force and Radius - so, if the Radius is straight along the X-axis and Force is along the Y, Torque can only be in the Z-axis.
To find out exactly where this vector will end up, we do one of the above operations, preferrably breaking both vectors up into i j and k components and multiplying them with one another. (Anark points out that this is called Cyclic Permutation.) It is imperative to remember that if an i term is multiplied with a j term it turns into a k term, and if a j term is multiplied with an i it becomes -k.
A simple way to remember this is that if we go forward in ijk, it stays positive, but if we go backwards it turns negative:
i * j = k
j * k = i
i * k = j
j * i = -k
k * j = -i
k * i = -j
To save time on math finals and real-world applications (there are such things) we can simplify in most cases. Something big and ugly like:
(3 cos wti + 3 sin wtj + 4wtk) x (-3 w sin wti + 3w cos wtj + 4wtk)
w without bold denotes angle measure in this problem. What we're actually doing here, is finding the cross product between r and r' -- angular momentum.
Can be simplified by noticing that:
u = cos wti + sin wtj,
v = -sin wti + cos wtj,
w = k
form a right-hand rule set of coordinate axes -- their cross product works just like that of i, j and k
The cross product then, is:
(3u + 4tww) x (3wv + 4ww) =
9wu x v + 12wu x w + 12tww x v =
9ww - 12wv - 12twu