Question
A unit vector perpendicular to the plane passing through the points
whose position vectors are $$\overrightarrow i - \overrightarrow j + 2\overrightarrow k ,\,2\overrightarrow i - \overrightarrow k $$ and $$2\overrightarrow j + \overrightarrow k $$ is :
A.
$$2\overrightarrow i + \overrightarrow j + \overrightarrow k $$
B.
$$\frac{1}{{\sqrt 6 }}\left( {2\overrightarrow i + \overrightarrow j + \overrightarrow k } \right)$$
C.
$$\frac{1}{{\sqrt 6 }}\left( {\overrightarrow i + 2\overrightarrow j + \overrightarrow k } \right)$$
D.
none of these
Answer :
$$\frac{1}{{\sqrt 6 }}\left( {2\overrightarrow i + \overrightarrow j + \overrightarrow k } \right)$$
Solution :
Let the points be $$A,\,B,\,C.$$ Then $$\overrightarrow {OA} = \overrightarrow i - \overrightarrow j + 2\overrightarrow k ,\,\overrightarrow {OB} = 2\overrightarrow i - \overrightarrow k ,\,\overrightarrow {OC} = 2\overrightarrow j + \overrightarrow k .$$
The required vector $$ = \frac{{\overrightarrow {AB} \times \overrightarrow {AC} }}{{\left| {\overrightarrow {AB} \times \overrightarrow {AC} } \right|}}.$$ Calculate its value.