Question

If a line lies in the octant OXYZ and it makes equal angle with axis then
A. $l = m = n = \dfrac{1}{{\sqrt 3 }}$
B. $l = m = n = \pm \dfrac{1}{{\sqrt 3 }}$
C. $l = m = n = - \dfrac{1}{{\sqrt 3 }}$
D. $l = m = n = \pm \dfrac{1}{{\sqrt 2 }}$

Answer 1

Hint:To find out the direction cosine when the angles made by the line with axes are known to be in a pattern. When the angles formed by the lines with axes are known, finding the direction cosine (DC) of the line requires taking the value of angle and squaring their sum equal to one.

Formula Used:
The line's direction cosines is given by:
$ l = \cos \alpha, m = \cos \beta, n = \cos \gamma$
Relation between l,m and n :
$ l^2 + m^2 + n^2 = 1$

Complete step-by-step solution:
We have given an octant OXYZ and all the lines make equal angles with the coordinate axis. Therefore we can say that,
\[\angle x = \angle{\text{ }}y = \angle{\text{ }}z\] ---(1)
Let,$l = \cos \alpha $, $m = \cos \beta $and $n = \cos \gamma $
Since, ${\cos ^2}\alpha + {\cos ^2}\beta + {\cos ^2}\gamma = 1$ ---(2)
Using equation (1) in equation (2)
${\cos ^2}\alpha + {\cos ^2}\alpha + {\cos ^2}\alpha = 1$
$3{\cos ^2}\alpha = 1$
${\cos ^2}\alpha = \dfrac{1}{3}$
$\cos \alpha = \pm \dfrac{1}{{\sqrt 3 }}$
Therefore we can say that $\cos \alpha = l = m = n = \pm \dfrac{1}{{\sqrt 3 }}$
Hence the correct option is B$\left( {l = m = n = \pm \dfrac{1}{{\sqrt 3 }}} \right)$ .

So, option B is correct.

Note: Neglect the negative sign. Since all three components of the octant OXYZ are positive, the line now forms equal angles with each of the axes. In three-dimensional geometry, the direction cosines of the line are the cosines of each of the angles the line makes with the x, y, and z axes, respectively. These direction cosines are typically represented by the letters l, m, and n, respectively. In this case we knew that angles formed by the line are equal so, after taking them as equal we just need to substitute the angle with one to simplify.