Let $\vec{a}=\alpha \hat{i}+\hat{j}+\beta \hat{k}$ and $\vec{b}=3 \hat{i}-5 \hat{j}+4 \hat{k}$ be two vectors, such that $\vec{a} \times \vec{b}=-\hat{i}+9 \hat{j}+12 \hat{k}$. Then the projection of $\vec{b}-2 \vec{a}$ on $\vec{b}+\vec{a}$ is equal to :

<p>$\overrightarrow a = \alpha \widehat i + \widehat j + \beta \widehat k$, $\overrightarrow b = 3\widehat i - 5\widehat j + 4\widehat k$</p> <p>$$\overrightarrow a \times \overrightarrow b = - \widehat i + 9\widehat j + 12\widehat k$$</p> <p>$$\left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr \alpha & 1 & \beta \cr 3 & { - 5} & 4 \cr } } \right| = - \widehat i + 9\widehat j + 12\widehat k$$</p> <p>$4 + 5\beta = - 1 \Rightarrow \beta = - 1$</p> <p>$- 5\alpha - 3 = 12 \Rightarrow \alpha = - 3$</p> <p>$$\overrightarrow b - 2\overrightarrow a = 3\widehat i - 5\widehat j + 4\widehat k - 2\left( { - 3\widehat i + \widehat j - \widehat k} \right)$$</p> <p>$$\overrightarrow b - 2\overrightarrow a = 9\widehat i - 7\widehat j + 6\widehat k$$</p> <p>$$\overrightarrow b + \overrightarrow a = \left( {3\widehat i - 5\widehat j + 4\widehat k} \right) + \left( { - 3\widehat i + \widehat j - \widehat k} \right)$$</p> <p>$\overrightarrow b + \overrightarrow a = - 4\widehat j + 3\widehat k$</p> <p>Projection of $\overrightarrow b - 2\overrightarrow a$ on $\overrightarrow b + \overrightarrow a$ is $$ = {{\left( {\overrightarrow b - 2\overrightarrow a } \right)\,.\,\left( {\overrightarrow b + \overrightarrow a } \right)} \over {\left| {\overrightarrow b + \overrightarrow a } \right|}}$$</p> <p>$= {{28 + 18} \over 5} = {{46} \over 5}$</p>