Given two vectors #vec u, vec v# we know that
#<< vec u, vec v >> = norm(vec u) xx norm(vec v) xx cos(hat(vec u,vec v))#
Solving for the angle #hat(vec u,vec v)# we have
#hat(vec u,vec v) = arccos(<< vec u, vec v >>/(norm(vec u) xx norm(vec v)))#
In our example
#vec u = {5,5,-1}#
#vec v = {4,-3,1}#
#<< vec u, vec v >> =5 xx 4+5 xx (-3)+(-1)xx1 = 4#
#norm vec u = sqrt(5^2+5^2+(-1)^2) = sqrt(51)#
#norm vec v = sqrt(4^2+(-3)^2+1^2) = sqrt(26)#
then
#hat(vec u,vec v) = arccos(4/sqrt(51 xx 26)) = 1.46073# [rad] #approx 83.70^@#
