Question
Under what condition do $$\left\langle {\frac{1}{{\sqrt 2 }},\,\frac{1}{2},\,k} \right\rangle $$ represent direction cosines of a line ?
A.
$$k = \frac{1}{2}$$
B.
$$k = - \frac{1}{2}$$
C.
$$k = \pm \frac{1}{2}$$
D.
$$k$$ can take any value
Answer :
$$k = \pm \frac{1}{2}$$
Solution :
For $$\left( {\frac{1}{{\sqrt 2 }},\,\frac{1}{2},\,k} \right)$$ to represent direction cosines, we should have
$$\eqalign{
& {\left( {\frac{1}{{\sqrt 2 }}} \right)^2} + {\left( {\frac{1}{2}} \right)^2} + {k^2} = 1\,\,{\text{or, }}\frac{1}{2} + \frac{1}{4} + {k^2} + 1 \cr
& \Rightarrow k = \pm \frac{1}{2} \cr} $$