Direction Cosines

Direction cosines l, m, n of a vector are the cosines of the angles it makes with the x, y and z axes, and they always satisfy the fundamental relation l^2 + m^2 + n^2 = 1. This normalization identity is the backbone of JEE Advanced direction-cosine problems. If the vector makes equal angles with all three axes, then l = m = n, and substituting into the identity gives 3 l^2 = 1, so l^2 = 1/3 and l = 1/sqrt(3) (taking the positive root for acute angles). Hence each direction cosine equals 1/sqrt(3). Option 1/3 mistakenly reports l^2 rather than l itself. Option 1/sqrt(2) would satisfy only a two-axis relation, 2 l^2 = 1, not all three. Option sqrt(3) exceeds one and is impossible since a cosine cannot surpass unity. The equal-angle line is the main diagonal of the coordinate cube, whose direction cosines are famously 1/sqrt(3) each. Plausibility check: summing the squares gives 3 times 1/3 = 1, satisfying the identity exactly, and each value lies in (0,1) as a valid cosine must.