Calling #vec u = {2,0,1,-4}# and #vec v = {2,3,0,1}#
we need a vector
# vec x = {a,b,c,d}#
such that
#<< vec u, vec x >> = 0#
#<< vec v, vec x >> = 0#
#norm (vec x) = 1#
Solving
#{
(2 a + c - 4 d = 0),
(2 a + 3 b + d = 0),
(a^2 + b^2 + c^2 + d^2 = 1)
:}#
for #a,b,c# we obtain
#((a = 1/7 (10 d - 3 sqrt[1 - 6 d^2])),( b =
1/7 (-9 d + 2 sqrt[1 - 6 d^2])), (c = 2/7 (4 d + 3 sqrt[1 - 6 d^2])))#
or
#((a = 1/7 (10 d + 3 sqrt[1 - 6 d^2])),( b =
1/7 (-9 d - 2 sqrt[1 - 6 d^2])), (c = 2/7 (4 d - 3 sqrt[1 - 6 d^2])))#
then if we choose #1-6d^2 ge 0# or #-1/sqrt(6) le d le 1/sqrt(6)# we will have solutions to this problem
