Question

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

Options

• \[13; \frac{12}{13}, \frac{4}{13}, \frac{3}{13}\]

• \[19; \frac{12}{19}, \frac{4}{19}, \frac{3}{19}\]

• \[11; \frac{12}{11}, \frac{14}{11}, \frac{3}{11}\]

•  none of these

Answer 1

 \[13; \frac{12}{13}, \frac{4}{13}, \frac{3}{13}\]  

If a line makes angles α, β 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 

\[\frac{12}{13}, \frac{4}{13}, \frac{1}{13}\]