The direction-cosines of the two lines are given by the equations
u l + v m + w n = 0    ....(1)
f m n + g n l + h l m = 0    ....(2)
From (1),  w n = -(u l +  v m),         
Putting this value of n in (2), we get,

∴    –f m (u l + v m)–g l(u l + v m) + h lm w = 0
∴    –u f l m  – v f m² – u g l² – v g l m + h l m w = 0
∴    – u g l² – (u f + v g – h w) Im – v f m² = 0
or     u g l² + (u f + v g –h w )l m + v f m² = 0
or                            ...(3)
which is a quadratic is 
Let  be its roots where l₁ , m₁ , n₁ and l₂, m₂, n₂ are the direction-cosines of the lines.
(i) The two lines will be perpendicular when
l₁ l₂ + m₁ m₂ + n₁ n₂  = 0    .....(4)
From (3),   

     

or                            (By symmetry)



Putting in (4), we see that lines are perpendicular
if  

i.e., if 
which is required condition. 

(ii) The lines are parallel if l= l₂, m₁= m₂, n₁ = n₂
i.e.,   if 
i.e.,    if the roots of (3) are equal
i.e.,    if disc, of (3) = 0
i.e.,    if (u f + v g – h w)² – 4 (u g) (v f) = 0
i.e.,    if (u f + v g – h w)² = 4 u v f g
i.e.,  if 
i.e.,  if 
i.e., if 
i.e., if 
i.e., if