Let a vector $\overrightarrow c$ be coplanar with the vectors $\overrightarrow a = - \widehat i + \widehat j + \widehat k$ and $\overrightarrow b = 2\widehat i + \widehat j - \widehat k$. If the vector $\overrightarrow c$ also satisfies the conditions $$\overrightarrow c \,.\,\left[ {\left( {\overrightarrow a + \overrightarrow b } \right) \times \left( {\overrightarrow a \times \overrightarrow b } \right)} \right] = - 42$$ and $$\left( {\overrightarrow c \times \left( {\overrightarrow a - \overrightarrow b } \right)} \right)\,.\,\widehat k = 3$$, then the value of $|\overrightarrow c {|^2}$ is equal to :

<p>Given,</p> <p>$\overrightarrow a = - \widehat i + \widehat j + \widehat k$</p> <p>$\overrightarrow b = 2\widehat i + \widehat j - \widehat k$</p> <p>and let $\overrightarrow c = x\widehat i + y\widehat j + z\widehat k$</p> <p>Now, $\overrightarrow a + \overrightarrow b = \widehat i + 2\widehat j$</p> <p>and $$\overrightarrow a \times \overrightarrow b = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr { - 1} & 1 & 1 \cr 2 & 1 & { - 1} \cr } } \right| = - 2\widehat i + \widehat j - 3\widehat k$$</p> <p>$\overrightarrow c$ is coplanar with $\overrightarrow a$ and $\overrightarrow b$</p> <p>$\therefore$ $\left[ {\overrightarrow c \,\overrightarrow a \,\overrightarrow b } \right] = 0$</p> <p>$$ \Rightarrow \left| {\matrix{ x & y & z \cr { - 1} & 1 & 1 \cr 2 & 1 & { - 1} \cr } } \right| = 0$$</p> <p>$\Rightarrow x( - 2) - y( - 1) + z( - 3) = 0$</p> <p>$\Rightarrow - 2x + y - 3z = 0$ ..... (1)</p> <p>Now, $$\left( {\overrightarrow a + \overrightarrow b } \right) \times \left( {\overrightarrow a \times \overrightarrow b } \right)$$</p> <p>$$ = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 1 & 2 & 0 \cr { - 2} & 1 & { - 3} \cr } } \right|$$</p> <p>$= - 6\widehat i + 3\widehat j + 5\widehat k$</p> <p>Given, $$\overrightarrow c \,.\,\left[ {\left( {\overrightarrow a + \overrightarrow b } \right) \times \left( {\overrightarrow a \times \overrightarrow b } \right)} \right] = - 42$$</p> <p>$$ \Rightarrow \left( {x\widehat i + y\widehat j + z\widehat k} \right)\,.\,\left( { - 6\widehat i + 3\widehat j + 5\widehat k} \right) = - 42$$</p> <p>$\Rightarrow - 6x + 3y + 5z = - 42$ ...... (2)</p> <p>Now, $$\overrightarrow a - \overrightarrow b = \left( { - \widehat i + \widehat j + \widehat k} \right) - \left( {2\widehat i + \widehat j - \widehat k} \right)$$</p> <p>$= - 3\widehat i + 0\widehat j + 2\widehat k$</p> <p>$\therefore$ $$\overrightarrow c \times \left( {\overrightarrow a - \overrightarrow b } \right)$$</p> <p>$$ = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr x & y & z \cr { - 3} & 0 & 2 \cr } } \right|$$</p> <p>$= 2y\widehat i - \widehat j(2x + 3z) + 3y\widehat k$</p> <p>Given,</p> <p>$$\left( {\overrightarrow c \times \left( {\overrightarrow a - \overrightarrow b } \right)} \right)\,.\,\widehat k = 3$$</p> <p>$$ \Rightarrow \left( {2y\widehat i - \widehat j(2x + 3z) + 3y\widehat k} \right)\,.\,\widehat k = 3$$</p> <p>$\Rightarrow 3y = 3$</p> <p>$\Rightarrow y = 1$</p> <p>Putting value of $y = 1$ in equation (1) and (2) we get,</p> <p>$- 2x + 1 - 3z = 0$ ..... (3)</p> <p>and $- 6x + 3 + 5z = - 42$</p> <p>$\Rightarrow - 6x + 5z = - 45$ ..... (4)</p> <p>Solving (3) and (4), we get</p> <p>$x = 5$ and $z = - 3$</p> <p>$\therefore$ $\overrightarrow c = 5\widehat i + \widehat j - 3\widehat k$</p> <p>$$ \Rightarrow {\left| {\overrightarrow c } \right|^2} = {5^2} + {1^2} + {( - 3)^2} = 35$$</p>