Question
Let $$\overrightarrow a = 2\overrightarrow i - \overrightarrow j + \overrightarrow k ,\,\overrightarrow b = \overrightarrow i + 2\overrightarrow j - \overrightarrow k $$ and $$\overrightarrow c = \overrightarrow i + \overrightarrow j - 2\overrightarrow k .$$ A vector in the plane of $$\overrightarrow b $$ and $$\overrightarrow c $$ whose projection on $$\overrightarrow a $$ has the magnitude $$\sqrt {\frac{2}{3}} $$ is :
A.
$$2\overrightarrow i + 3\overrightarrow j - 3\overrightarrow k $$
B.
$$2\overrightarrow i + 3\overrightarrow j + 3\overrightarrow k $$
C.
$$ - 2\overrightarrow i - \overrightarrow j + 5\overrightarrow k $$
D.
$$2\overrightarrow i + \overrightarrow j + 5\overrightarrow k $$
Answer :
$$ - 2\overrightarrow i - \overrightarrow j + 5\overrightarrow k $$
Solution :
Let $$\overrightarrow p = \lambda \overrightarrow b + \mu \overrightarrow c $$
The projection of $$\overrightarrow p $$ on $$\overrightarrow a = \frac{{\overrightarrow p .\overrightarrow a }}{{\left| {\overrightarrow a } \right|}} = \sqrt {\frac{2}{3}} ......\left( 1 \right)$$
$$\eqalign{
& \because \,\overrightarrow p = \lambda \left( {\overrightarrow i + 2\overrightarrow j - \overrightarrow k } \right) + \mu \left( {\overrightarrow i + \overrightarrow j - 2\overrightarrow k } \right) \cr
& \therefore \,\overrightarrow p .\overrightarrow a = 2\left( {\lambda + \mu } \right) - 1\left( {2\lambda + \mu } \right) + 1\left( { - \lambda - 2\mu } \right) = - \lambda - \mu \cr
& \therefore \,\left( 1 \right){\text{gives}}\frac{{ - \lambda - \mu }}{{\sqrt {{2^2} + {1^2} + {1^2}} }} = \sqrt {\frac{2}{3}} \,\,\,{\text{or }}\lambda + \mu = - 2 \cr
& \therefore \overrightarrow p = \lambda \left( {\overrightarrow i + 2\overrightarrow j - \overrightarrow k } \right) + \mu \left( {\overrightarrow i + \overrightarrow j - 2\overrightarrow k } \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {\lambda + \mu } \right)\overrightarrow i + \left( {2\lambda + \mu } \right)\overrightarrow j - \left( {\lambda + 2\mu } \right)\overrightarrow k \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - 2\overrightarrow i + \left( {\lambda - 2} \right)\overrightarrow j + \left( {\lambda + 4} \right)\overrightarrow k , \cr} $$
where $$\lambda $$ is a scalar parameter.
When $$\lambda = 1,\,\overrightarrow p = - 2\overrightarrow i - \overrightarrow j + 5\overrightarrow k .$$ Other options hold for no real $$\lambda .$$