Let $\overrightarrow a = 4\widehat i + 3\widehat j$ and $\overrightarrow b = 3\widehat i - 4\widehat j + 5\widehat k$. If $\overrightarrow c$ is a vector such that $$\overrightarrow c .\left( {\overrightarrow a \times \overrightarrow b } \right) + 25 = 0,\overrightarrow c \,.(\widehat i + \widehat j + \widehat k) = 4$$, and projection of $\overrightarrow c$ on $\overrightarrow a$ is 1, then the projection of $\overrightarrow c$ on $\overrightarrow b$ equals :

<p>$$[\matrix{ {\overrightarrow c } & {\overrightarrow a } & {\overrightarrow b } \cr } ] = - 25$$</p> <p>Let $\overrightarrow c = l\widehat i + n\widehat j + n\widehat k$</p> <p>$$\left| {\matrix{ l & m & n \cr 4 & 3 & 0 \cr 3 & { - 4} & 5 \cr } } \right| = - 25$$</p> <p>$\Rightarrow 3l - 4m - 5n = - 5$ ..... (i)</p> <p>$\overrightarrow c \,.\,(\widehat i + \widehat j + \widehat k) = 4$</p> <p>$\Rightarrow l + m + n = 4$ ..... (ii)</p> <p>$${{\overrightarrow c \,.\,\overrightarrow a } \over {|\overrightarrow a |}} = 1 \Rightarrow \overrightarrow c \,.\,\overrightarrow a = 5$$</p> <p>$\Rightarrow 4l + 3m = 5$ ...... (iii)</p> <p>Using (i), (ii) and (iii)</p> <p>$l = 2,m = - 1,n = 3$</p> <p>Now, $${{\overrightarrow c \,.\,\overrightarrow b } \over {|\overrightarrow b |}} = {{25} \over {5\sqrt 2 }} = {5 \over {\sqrt 2 }}$$</p> <p>$\therefore$ Option (2) is correct.</p>