Let $\overrightarrow a$ be a vector which is perpendicular to the vector $3\widehat i + {1 \over 2}\widehat j + 2\widehat k$. If $$\overrightarrow a \times \left( {2\widehat i + \widehat k} \right) = 2\widehat i - 13\widehat j - 4\widehat k$$, then the projection of the vector $\overrightarrow a$ on the vector $2\widehat i + 2\widehat j + \widehat k$ is :

<p>Let $\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k$</p> <p>and $$\overrightarrow a \,.\,\left( {3\widehat i - {1 \over 2}\widehat j + 2\widehat k} \right) = 0 \Rightarrow 3{a_1} + {{{a_2}} \over 2} + 2{a_3} = 0$$ ..... (i)</p> <p>and $$\overrightarrow a \times (2\widehat i + \widehat k) = 2\widehat i - 13\widehat j - 4\widehat k$$</p> <p>$$ \Rightarrow {a_2}\widehat i + (2{a_3} - {a_1})\widehat j - 2{a_2}\widehat k = 2\widehat i - 13\widehat j - 4\widehat k$$</p> <p>$\therefore$ ${a_2} = 2$ ..... (ii)</p> <p>and ${a_1} - 2{a_3} = 13$ ..... (iii)</p> <p>From eq. (i) and (iii) : ${a_1} = 3$ and ${a_3} = - 5$</p> <p>$\therefore$ $\overrightarrow a = 3\widehat i + 2\widehat j - 5\widehat k$</p> <p>$\therefore$ projection of $\overrightarrow a$ on $2\widehat i + 2\widehat j + \widehat k = {{6 + 4 - 5} \over 3} = {5 \over 3}$</p>