Given, pl + qm + nr = 0 ...........1

and al² + bm² + cn² = 0 .........2

From equation 1, we get

l = -(qm + nr)/p

Put value of l in equation 2, we get

=> a{-(qm + nr)/p}² + bm² + cn² = 0

=> a{(qm + nr)/p}² + bm² + cn² = 0

=> a(qm + nr)² /p² + bm² + cn² = 0

=> a(qm + nr)² + bm² p² + cn² p² = 0

=> a(q² m² + n² r² + 2qr * mn) + bm² p² + cn² p² = 0

=> aq² m² + an² r² + 2aqr * mn + bm² p² + cn² p² = 0

=> aq² m² + bm² p² + an² r² + cn² p² + 2aqr * mn = 0

=> m² (aq² + bp² ) + n² (ar² + cp² ) + 2aqr * mn = 0

=> (m² /n² )*(aq² + bp² ) + (ar² + cp² ) + 2aqr * m/n = 0         {divide by n² }

=> (aq² + bp² ) * (m/n)² + 2aqr * (m/n) + (ar² + cp² ) = 0

=> m₁ m₂ /n₁ n₂ = (ar² + cp² )/(aq² + bp² )

=> m₁ m₂ /(ar² + cp² ) = n₁ n₂ /(aq² + bp² )  ..................3

Again from equation 1, we get

n = -(pl + qm)/r

Put value of n in equation 2, we get 

=> al² + bm² + c{-(pl + qm)/r}² = 0

=> al² + bm² + c(pl + qm)² /r² = 0

=> al² r² + bm² r² + c(pl + qm)² = 0

=> al² r² + bm² r² + c(p² l² + q² m² + 2pq * lm) = 0

=> al² r² + bm² r² + cp² l² + cq² m² + 2pqc * lm = 0

=> al² r² + cp² l² + bm² r² +  cq² m² + 2pqc * lm = 0

=> (ar² + cp² )l² + (br² +  cq² )m² + 2pqc * lm = 0

=> (ar² + cp² ) * (l² /m² )+ (br² +  cq² ) + 2pqc * (l/m) = 0

=> (ar² + cp² ) * (l/m)² + 2pqc * (l/m) + (br² +  cq² ) = 0

=> l₁ l₂ /m₁ m₂ = (br² +  cq² )/(ar² + cp² )

=> l₁ l₂ /(br² +  cq² ) = m₁ m₂ /(ar² + cp² )  .........4

From equation 3 and 4, we get

l₁ l₂ /(br² +  cq² ) = m₁ m₂ /(ar² + cp² ) = n₁ n₂ /(aq² + bp² ) = k (say)

=> l₁ l₂  = k(br² +  cq² ), m₁ m₂ = k(ar² + cp² ), n₁ n₂ = k(aq² + bp² )

Now, two lines are perpendicular if

=> l₁ l₂ + m₁ m₂ + n₁ n₂ = 0

=> k(br² +  cq² ) + k(ar² + cp² ) + k(aq² + bp² ) = 0

=> k(br² +  cq² + ar² + cp² + aq² + bp² ) = 0

=> br² +  cq² + ar² + cp² + aq² + bp² = 0

=> p² (b + c) + q² (a + c) + r² (a + b) = 0

This is the required condition.

Similarly, we can show that if two lines are parallel then,

p² /a + q² /b + c² /a = 0