= 0
Dividing both sides by m², we get,
                 ...(3)
which is a quadratic in 
Let l₁, m₁, n₁ ; l₂, m₂, n₂ be the direction-cosines of the two lines. Then  are the roots of the equation (3).
(i) The lines will be perpendicular when 
       l₁ l₂ + m₁ m₂ + n₁ n₂ = 0    ....(4)
From (3), 

∴  l₁ l₂ = k (b w² + c v²), m₁ m₂ = k (c u² + a w²), n₁ n₂ = k (a v² + b u²)
∴  l₁ l₂ + m₁ m₂ + n₁ n₂ = k [b w²'+ cv² + c u² + a w'² + a v² + b u²]
∴  lines are perpendicular
If k (b w² + cv² + c u² + a w² + a v² + b u²] = 0    [∵ of (4)]
i.e., if b w² + c v² + c u² + a w² + a v² + b u² = 0
i.e., if u² (b + c) + v² (c + a) + w² (a + b) = 0

(ii) The lines are parallel
if l₁ = l₂, m₁ = m₂, n₁ = n₂
i.e.,  if 
i.e.,    if equation (3) has equal roots    
i.e.,    if disc = 0
i.e.,    if 4 c² u² v² – 4 (a w² + c u²) (b w² + c v²) = 0
i.e.,    lf c² u² v² – a bw⁴ – acv² w² –bcu² W – c² u² v²= 0
i.e.,    if – a b w⁴ – a c v² w² – b c u² w² = 0
i.e.,    if a b w² + a c v² + b c u² = 0    [Dividing by – w²]
i.e,  if 

i.e., if