The lines with direction cosines 12/17, -3/17, -4/17 and 4/17, 12/17,…

The lines with direction cosines l₁, m₁, n₁ and l₂, m₂,n₂  are perpendicular to each other if

l₁.l₂+ m₁.m₂+ n₁.n₂ = 0

We have the lines with direction cosines 12/17, -3/17, -4/17 and 4/17, 12/17, 3/17

so  l₁.l₂+ m₁.m₂+ n₁.n2 = (12/17).(4/17) + (-3/17).(12/17) + (-4/17).(3/17)

⇒  l₁.l₂+ m₁.m₂+ n₁.n2 = (48/289) -(36/289) - (12/289)

⇒  l₁.l₂+ m₁.m₂+ n₁.n₂ = 0

Therefore the line are perpendicular.