Cross Product

a = {a1, a2, a3}

b = {b1, b2, b3}

Cross3D[a_, b_] := {
        a[[2]] 
            b[[3]] - 
          a[[3]] 
            b[[2]], 
        a[[3]] 
            b[[1]] - 
          a[[1]] 
            b[[3]], 
        a[[1]] 
            b[[2]] - 
          a[[2]] 
            b[[1]]}

Cross3D[a, a]

Cross3D[b, b]

Cross3D[a, b]

c = {c1, c2, c3}

triple = Cross3D[a, Cross3D[b, c]];

Expand[%]

guess = a.c b - a.b c;

Expand[%]

Expand[triple - guess]

r = {x, y, z}

EField = {Ex, Ey, Ez}

BField = {Bx, By, Bz}

Cross3D[r, Cross3D[EField, BField]];

Expand[%]

r.Cross3D[EField, BField];

Expand[%]