What are cross products?
When you encounter vectors in
#vec(u) * vec(v) = u_1v_1 + u_2v_2 + u_3v_3#
#vec(u) xx vec(v) = < u_2v_3-u_3v_2, color(white)(.)u_3v_1-u_1v_3, color(white)(.)u_1v_2-u_2v_1 >#
This is sometimes described in terms of a determinant of a
#vec(u) xx vec(v) = abs((hat(i), hat(j), hat(k)), (u_1, u_2, u_3), (v_1, v_2, v_3))#
How about division ?
Neither dot product nor cross product allow division of vectors. To find how to divide vectors you can look at the quaternions. The quaternions form a
Anyway, we can say that a quaternion can be written as a combination of a scalar part and vector part, with arithmetic defined by:
#(r_1, vec(v_1)) + (r_2, vec(v_2)) = (r_1 + r_2, vec(v_1) + vec(v_2))#
#(r_1, vec(v_1)) * (r_2, vec(v_2)) = (r_1 r_2 - vec(v_1) * vec(v_2), r_1 vec(v_2) + r_2 vec(v_1) + vec(v_1) xx vec(v_2))#
For a very interesting related talk, watch this...