Direction Cosines From Two Points

Question

A directed line segment joins the point (3, 4, 5) to the point (5, 6, 7), so what are its direction cosines?

RankGuru · Accepted Answer

The relevant method is forming direction ratios from coordinate differences, then normalizing to get direction cosines: the ratios are (x_2 - x_1, y_2 - y_1, z_2 - z_1), and dividing by the segment length yields the cosines. This two-point procedure is a basic JEE Advanced building block. The differences are (5-3, 6-4, 7-5) = (2, 2, 2). The length is \sqrt{2^2 + 2^2 + 2^2} = \sqrt{12} = 2\sqrt{3}. Dividing each ratio by 2\sqrt{3} gives (2/(2\sqrt{3}), 2/(2\sqrt{3}), 2/(2\sqrt{3})) = (1/\sqrt{3}, 1/\sqrt{3}, 1/\sqrt{3}). Option (2, 2, 2) lists direction ratios, not cosines, and violates l^2+m^2+n^2 = 1. Option (1/3, 1/3, 1/3) divides by the wrong magnitude. Option (2/\sqrt{3}, ...) forgets the factor of 2 in the length. This applies the direction-cosine normalization theorem. Plausibility check: (1/\sqrt{3})^2 summed three times equals 3 \cdot 1/3 = 1, satisfying the unit-sum identity for direction cosines.

Direction Cosines From Two Points

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