Question

The projections of a line segment on the coordinate axes are 12, 4, 3. The direction cosine of the line are:

• A. $-\frac{12}{13},-\frac{4}{13},\frac{3}{13}$
• B. $\frac{12}{13},-\frac{4}{13},\frac{3}{13}$
• C. $\frac{12}{13},\frac{4}{13},\frac{3}{13}$
• D. none of these

Answer 1

Answer: (C)

Let direction cosine of the line are l, m, n
where Given, $\cos \theta =\frac{D{C}^{\prime }s}{r}$
$\Rightarrow ,D{C}^{\prime }s=r\cos \theta =rl$
$\Rightarrow rl=12$ .....(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 $\frac{12}{13},\frac{4}{13},\frac{3}{13}$