math vectors- angle between vectors

u=( -5 h k ) is orthogonal with both v= ( 1 3 2 ) and w= ( -2 1 4 )
so
u.v=0
u.w=0
from
u.v=0
-5+3h+2k=0...(1)
from
u.w=0
10+h+4k=0...(2)
from (2)
h=-4k-10
sub into (1)
-5+3h+2k=0
-5+3(-4k-10)+2k=0
-5-12k-30+2k=0
-10k=35
k=-3.5
h=-4k-10=4

u=(-5,h,k)
When u．v=0 and u．w=0
u will be orthogonal with v and w
(-5)(1)+3h+2k=0......(1)
(-5)(-2)+h+4k=0......(2)
(1):3h+2k=5......(3)
(2):h+4k=-10......(4)
(4)-(3)*2:-5h=-20
h=4
Put h=4 into (3)
12+2k=5
k=-3.5