Question

If 1,2,1 are the direction ratios of a line then which of the following could be direction cosines of this line?

• A.
• B.
• C.
• D. None of these

Answer 1

The correct option is B

We know that the direction ratios are proportional to direction cosines.
I.e. l = k.a, m = k.b , n = k.c
where k is any constant, l,m & n are direction cosines and a,b & c are direction ratios.
We also know that $l^2+m^2+n^2=1$
So,$(K.a{)}^2+(K.b{)}^2+(K.c{)}^2=1$
$K^2(a^2+b^2+c^2)=1$
$k^2=\frac{1}{(a^2+b^2+c^2)}$
Or, $K=\pm \sqrt{\frac{1}{(a^2+b^2+c^2)}}$
In the question given direction ratios or a,b,c are 1, 2, 1.
So, $K=\pm \sqrt{\frac{1}{\left(\right(1{)}^2+(2{)}^2+(1{)}^2)}}$
Or, $K=\pm \sqrt{\frac{1}{6}}$
So the direction cosines will be -
I=1.k
or,$l=\sqrt{\frac{1}{6}}\left(ontakingkas\sqrt{\frac{1}{6}}\right)$
Similarly ,$m=2.\sqrt{\frac{1}{6}}$ $\&$ n=$\sqrt{\frac{1}{6}}$
Also,-$\sqrt{\frac{1}{6}},-2.\sqrt{\frac{1}{6}},-\sqrt{\frac{1}{6}}$ on taking k as $-\sqrt{\frac{1}{6}}$