Direction Cosines And Direction Ratios

Question

A line makes equal acute angles with the three coordinate axes, so what is the value of each of its direction cosines?

RankGuru · Accepted Answer

The governing identity here is the fundamental relation for direction cosines: if l, m, n are the cosines of the angles a line makes with the x, y and z axes, then l^2 + m^2 + n^2 = 1. This constraint always holds for any line in space, which is the core JEE Advanced fact being probed. Since the line makes equal acute angles with all three axes, l = m = n. Substituting into the identity gives 3l^2 = 1, so l^2 = 1/3 and l = \pm 1/\sqrt{3}. Because the angles are acute, the cosines are positive, hence each direction cosine equals 1/\sqrt{3}. Option 1/3 is wrong because that is the value of l^2, not l. Option \sqrt{3} is impossible since a cosine cannot exceed 1. Option 1/\sqrt{2} would force l^2 + m^2 + n^2 = 3/2 
eq 1, violating the identity. This uses the standard direction-cosine normalization theorem. Plausibility check: 1/\sqrt{3} \approx 0.577 corresponds to an angle of about 54.7 degrees, which is acute and consistent with three equal positive cosines summing in square to one.

Direction Cosines And Direction Ratios

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