Question
If $$\overrightarrow a = \overrightarrow i + \overrightarrow j ,\,\overrightarrow b = 2\overrightarrow j - \overrightarrow k $$
and $$\overrightarrow r \times \overrightarrow a = \overrightarrow b \times \overrightarrow a ,\,\overrightarrow r \times \overrightarrow b = \overrightarrow a \times \overrightarrow b $$ then $$\frac{{\overrightarrow r }}{{\left| {\overrightarrow r } \right|}}$$ is equal to :
A.
$$\frac{1}{{\sqrt {11} }}\left( {\overrightarrow i + 3\overrightarrow j - \overrightarrow k } \right)$$
B.
$$\frac{1}{{\sqrt {11} }}\left( {\overrightarrow i - 3\overrightarrow j + \overrightarrow k } \right)$$
C.
$$\frac{1}{{\sqrt 3 }}\left( {\overrightarrow i - \overrightarrow j + \overrightarrow k } \right)$$
D.
none of these
Answer :
$$\frac{1}{{\sqrt {11} }}\left( {\overrightarrow i + 3\overrightarrow j - \overrightarrow k } \right)$$
Solution :
$$\eqalign{
& {\text{Here, }}\overrightarrow r \times \overrightarrow a + \overrightarrow r \times \overrightarrow b = 0\,\,\,{\text{or }}\overrightarrow r \times \left( {\overrightarrow a + \overrightarrow b } \right) = 0\,\, \cr
& \therefore \overrightarrow r ||\left( {\overrightarrow a + \overrightarrow b } \right) \cr
& \therefore \,\overrightarrow r = t\left( {\overrightarrow a + \overrightarrow b } \right) = t\left( {\overrightarrow i + 3\overrightarrow j - \overrightarrow k } \right) \cr
& \therefore \,\left| {\overrightarrow r } \right| = t.\sqrt {{1^2} + {3^2} + {1^2}} = t.\sqrt {11} \cr
& \therefore \,\frac{{\overrightarrow r }}{{\left| {\overrightarrow r } \right|}} = \frac{1}{{\sqrt {11} }}\left( {\overrightarrow i + 3\overrightarrow j - \overrightarrow k } \right) \cr} $$