The projections of a line segment on X, Y and Z axes are 12, 4 and 3 respectively. The length and direction cosines of the line segment are

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13; $\frac{12}{13}$, $\frac{4}{13}$, $\frac{3}{13}$ If a line makes angles $\alpha$, $\beta$ and γ with the axes, then \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 . . . (1) Let r be the length of the line segment. Then, r \cos \alpha = 12, r \cos \beta = 4, r \cos \gamma = 3 . . . (2) \Rightarrow \left( r \cos \alpha \right)^2 + \left( r \cos \beta \right)^2 + \left( r \cos \gamma \right)^2 = {12}^2 + 4^2 + 3^2 \Rightarrow r^2 \left( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma \right) = 169 \Rightarrow r^2 \left( 1 \right) = 169 \left[ \text { From }\left( 1 \right) \right] \Rightarrow r = $\sqrt$(169) \Rightarrow r = $\pm$ 13 \Rightarrow r = 13 ( \text{ Since length cannot be negative } ) Substituting r = 13 in (2), we get , \cos \alpha = $\frac{12}{13}$, \cos \beta = $\frac{4}{13}$, \cos \gamma = $\frac{1}{13}$ Thus, the direction cosines of the line are $\frac{12}{13}$, $\frac{4}{13}$, $\frac{1}{13}$

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