The angle between the lines whose direction cosines satisfy the equations l+m+n=0 and l2=m2+n2 is π6π2π3π4

The direction cosines of a line are the cosines of the angles that the line makes with the positive directions of the coordinates axes. They are usually denoted by l, m, n.

Given the equations l + m + n = 0 and l^2 = m^2 + n^2, we can find the angle between the lines using the formula for the dot product of two vectors:

cos θ = (l1l2 + m1m2 + n1*n2) / sqrt[(l1^2 + m1^2 + n1^2) * (l2^2 + m2^2 + n2^2)]

For the first line, we have l1 = l, m1 = m, n1 = n. For the second line, we have l2 = -l, m2 = -m, n2 = -n (because of the equation l + m + n = 0).

Substituting these values into the formula, we get:

cos θ = (-ll - mm - n*n) / sqrt[(l^2 + m^2 + n^2) * (l^2 + m^2 + n^2)] = -1 / sqrt[2] = -1 / sqrt[2]

Therefore, θ = π/4.