Hi, very nice couple of articles. I love this subject. I have been pursuing higher dimensional math in my free time for the last few years. I have made some discoveries on my own, too. I would like to share something with you that might assist your learning…

The cross product in R3

The cross product in R3 of two vectors (a1,a2,a3) & (b1,b2,b3) is the determinant of the matrix shown above, which yields a third vector, let’s say, (c1,c2,c3)… which I’m sure you know is a vector that is orthogonal to the first two vectors.

The same computation can be extended to higher dimensions, as well! Here is the formula for calculating an orthogonal vector in R4 when given three vectors as input (a1,a2,a3,a4), (b1,b2,b3,b4) & (c1,c2,c3,c4)

… very useful!

The i, j, k, and w terms are just the basis vectors

i = (1,0,0,0)

j = (0,1,0,0)

k = (0,0,1,0)

w = (0,0,0,1)

If you are interested I have a couple more insights to share…

Thank you!