Question

The number of lines making equal angles with the coordinate axes in three dimensional geometry is equal to

• A. 3
• B. 4
• C. 2
• D. None of these

Answer 1

Answer: (D)

Let l, m and n be the direction cosines of a line.
Since, the line is equally inclined with OX, OY and OZ.
$\therefore$ $l=m-n$ $(\because \cos \alpha =\cos \beta =\cos \gamma )$
Now, $l^2+m^2+n^2=1\Rightarrow 3l^2=1$
$\Rightarrow$ $l^2=1/3$
$\Rightarrow$ $l=\pm \frac{1}{\sqrt{3}}$
Hence, the direction cosines of given line are
$\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}},$
Since, $+ve$ and $-ve$
signs can be arranged at three places in
$2\times 2\times 2=8$ ways.
Therefore, there are eight lines which are equally inclined with the coordinate axis.