Question
What are the direction cosines of a line which is equally inclined to the positive directions of the axes ?
A.
$$\left\langle {\frac{1}{{\sqrt 3 }},\,\frac{1}{{\sqrt 3 }},\,\frac{1}{{\sqrt 3 }}} \right\rangle $$
B.
$$\left\langle { - \frac{1}{{\sqrt 3 }},\,\frac{1}{{\sqrt 3 }},\,\frac{1}{{\sqrt 3 }}} \right\rangle $$
C.
$$\left\langle { - \frac{1}{{\sqrt 3 }},\, - \frac{1}{{\sqrt 3 }},\,\frac{1}{{\sqrt 3 }}} \right\rangle $$
D.
$$\left\langle {\frac{1}{3},\,\frac{1}{3},\,\frac{1}{3}} \right\rangle $$
Answer :
$$\left\langle {\frac{1}{{\sqrt 3 }},\,\frac{1}{{\sqrt 3 }},\,\frac{1}{{\sqrt 3 }}} \right\rangle $$
Solution :
Let $$\ell ,\,m,\,n$$ are the direction cosines of a line that is inclined equally at $$\alpha $$ to the +ve direction of axes.
Now, $$\ell = \cos \,\alpha ,\,m = \cos \,\alpha ,\,n = \cos \,\alpha $$
$$\eqalign{
& {\text{Also, }}{\ell ^2} + {m^2} + {n^2} = 1 \cr
& \Rightarrow 3\,{\cos ^2}\alpha = 1 \cr
& \Rightarrow \cos \,\alpha = \frac{1}{{\sqrt 3 }} \cr} $$
$$\therefore $$ dc’s of the line are : $$\left\langle {\frac{1}{{\sqrt 3 }},\,\frac{1}{{\sqrt 3 }},\,\frac{1}{{\sqrt 3 }}} \right\rangle $$