Question
If $$\overrightarrow r = \left( {\hat i + 2\hat j + 3\hat k} \right) + \lambda \left( {\hat i - \hat j + \hat k} \right)$$ and $$\overrightarrow r = \left( {\hat i + 2\hat j + 3\hat k} \right) + \mu \left( {\hat i + \hat j - \hat k} \right)$$ are two lines, then the equation of acute angle bisector of two lines is :
A.
$$\overrightarrow r = \left( {\hat i + 2\hat j + 3\hat k} \right) + t\left( {\hat j - \hat k} \right)$$
B.
$$\overrightarrow r = \left( {\hat i + 2\hat j + 3\hat k} \right) + t\left( {2\hat i} \right)$$
C.
$$\overrightarrow r = \left( {\hat i + 2\hat j + 3\hat k} \right) + t\left( {\hat j + \hat k} \right)$$
D.
None of these
Answer :
$$\overrightarrow r = \left( {\hat i + 2\hat j + 3\hat k} \right) + t\left( {\hat j - \hat k} \right)$$
Solution :
Lines are $$\overrightarrow r = \left( {\hat i + 2\hat j + 3\hat k} \right) + \lambda \left( {\hat i - \hat j + \hat k} \right)$$ and $$\overrightarrow r = \left( {\hat i + 2\hat j + 3\hat k} \right) + \mu \left( {\hat i + \hat j - \hat k} \right)$$
along vectors $$\left( {\hat i - \hat j + \hat k} \right)$$ and $$\left( {\hat i + \hat j - \hat k} \right)$$ respectively.
Angle between two lines
$$\eqalign{
& = {\cos ^{ - 1}}\left( {\frac{{\left( 1 \right) \times \left( 1 \right) + \left( { - 1} \right) \times \left( 1 \right) + \left( 1 \right) \times \left( { - 1} \right)}}{{\sqrt 3 \,\sqrt 3 }}} \right) \cr
& = {\cos ^{ - 1}}\left( { - \frac{1}{{\sqrt 3 }}} \right) \cr} $$
Which is an obtuse angle.
$$\therefore $$ Vector along acute angle bisector
$$\eqalign{
& = \lambda \left[ {\frac{{\hat i - \hat j + \hat k}}{{\sqrt 3 }} - \frac{{\hat i + \hat j - \hat k}}{{\sqrt 3 }}} \right] \cr
& = \frac{{2\lambda }}{{\sqrt 3 }}\left( { - \hat j + \hat k} \right) \cr} $$
$$\therefore $$ Equation of acute angle bisector
$$ = \left( {\hat i + 2\hat j + 3\hat k} \right) + t\left( {\hat j - \hat k} \right)$$