If a line makes angles α , β and γ with the axes then,

cos²α + cos²β + cos²γ = 1 → (1)

Let ‘r’ be the length of the line segment.

Then,

rcosα = 12, rcosβ = 4, rcosγ = 3 → (2)

Now,

(rcosα)² + (rcosβ)² + (rcosγ)² = 12² + 4² + 3² → From (2)

r² (cos²α + cos²β + cos²γ) = 169

r² (1) = 169 → From (1)

r

r

r = 13 → (Since length cannot be negative)

Substituting r = 13 in (2) , we get

cosα , cosβ , cosγ

Thus, the direction cosines of the line are - , , .