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Q.
The projections of a line segment on the coordinate axes are 12, 4, 3. The direction cosine of the line are:
Three Dimensional Geometry
Solution:
Let direction cosine of the line are l, m, n
where Given, $\cos \theta = \frac{DC's}{r}$
$\Rightarrow , DC's = r \, \cos \theta = rl $
$\Rightarrow \, rl = 12$ .....(i)
similarly r m = 4 .... (ii)
and r n = 3 .... (iii)
Squaring and adding equation (i), (ii)
and (iii), we get
$r^2(l^2 + m^2 + n^2) = 12^2 + 4^2 + 3^2$
$= 144 + 16 + 9 = 169$
$\Rightarrow \, r^2 = 169 $ $( \because \, l^2 + m^2 + n^2 = 1)$
$\Rightarrow \, r = 13$
Now, $ l = \frac{\text{projection} \, \text{on} \, x - \text{axis}}{\text{length} \, \text{of} \, \text{line} \, \text{segment}} = \frac{12}{13} $
similarly, $m = \frac{4}{13} , n = \frac{3}{13}$
Hence, Direction cosine are $\frac{12}{13}, \frac{4}{13} , \frac{3}{13}$