Question

Direction cosines of the line bisecting the obtuse angle between lines whose direction ratios are $(-1,0,1)$ and $(0,1,-1),$ is

• A. $(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})$
• B. $(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$
• C. $(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})$
• D. $(-\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}})$

Answer 1

The correct option is B $(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$

Given : $D.r^{\prime }s$ of line $L_1=(a_1,b_1,c_1)=(-1,0,1)$
$\Rightarrow D.c^{\prime }s$ of $L_1=(l_1,m_1,n_1)=(-\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}})$
and $D.r^{\prime }s$ of line $L_2=(a_2,b_2,c_2)=(0,1,-1)$
$\Rightarrow D.c^{\prime }s$ of $L_2=(l_2,m_2,n_2)=(0,\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})$
As, $l_1{l}_2+m_1{m}_2+n_1{n}_2<0$
So,
$D.r^{\prime }s$ of obtuse angle bisector is : $(l_1+l_2,m_1+m_2,n_1+n_2)=(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$
which is also direction cosine as $\sum (l_1+l_2{)}^2=1$