Find the d.c’s of a line that makes equal angels with the axes , and find number of such lines.

The direction cosines (d.c's) of a line that makes equal angles with the axes are all equal to 1/√3 or -1/√3. This is because the direction cosines are the cosines of the angles that the line makes with the x, y, and z axes, and if all these angles are equal, their cosines are also equal.

The direction cosines are given by the formula:

l = cos α = a/√(a² + b² + c²) m = cos β = b/√(a² + b² + c²) n = cos γ = c/√(a² + b² + c²)

where α, β, γ are the angles the line makes with the x, y, and z axes, and a, b, c are the direction ratios of the line. If the line makes equal angles with the axes, then α = β = γ, and a = b = c.

Substituting these values into the formula gives:

l = m = n = 1/√3 or -1/√3

The number of such lines is four. These are the lines in the positive and negative directions along the line x = y = z in 3D space.