Let → a = α ^ i + ^ j + β ^ k and → b = 3 ^ i - 5 ^ j + 4 ^ k be...

Let \overset{\to}{a} = \alpha\overset{^}{i} + \overset{^}{j} + \beta\overset{^}{k}, \overset{\to}{b} = 3\overset{^}{i} - 5\overset{^}{j} + 4\overset{^}{k}

Now finding \overset{\to}{a} \times \overset{\to}{b} = - \overset{^}{i} + 9\overset{^}{j} + 12\overset{^}{k}

= \left|\begin{cases}\overset{^}{i} & \overset{^}{j} & \overset{^}{k} \\ \alpha & 1 & \beta \\ 3 & - 5 & 4\end{cases}\right|

⇒ \left(4 + 5\beta\right)\overset{^}{i} + \left(3\beta - 4\alpha\right)\overset{^}{j} + \left(- 5\alpha - 3\right)\overset{^}{k}

= - \overset{^}{i} + 9\overset{^}{j} + 12\overset{^}{k}

Now on comparing side we get,

$4 + 5\beta = - 1, 3\beta - 4\alpha = 9, - 5\alpha - 3 = 12$

$\beta = - 1, \alpha = - 3$

So, \overset{\to}{a} = - 3\overset{^}{i} + \overset{^}{j} - \overset{^}{k} \text{ and } \overset{\to}{b} = 3\overset{^}{i} - 5\overset{^}{j} + 4\overset{^}{k}

So,\overset{\to}{a} + \overset{\to}{b} = - 4\overset{^}{j} + 3\overset{^}{k}

Now finding ${\left|\overset{\to}{a}\right|}^{2} = 11 \text{ and } {\left|\overset{\to}{b}\right|}^{2} = 50$

And $\overset{\to}{a} \cdot \overset{\to}{b} = - 9 + \left(- 5\right) - 4 = - 18$

Now projection of \left(\overset{\to}{b} - 2\overset{\to}{a}\right)$on$\overset{\to}{a} + \overset{\to}{b}$will be </p><p>$= \dfrac{\left(\overset{\to}{b} - 2\overset{\to}{a}\right) \cdot \left(\overset{\to}{a} + \overset{\to}{b}\right)}{\left|\overset{\to}{a} + \overset{\to}{b}\right|}$</p><p>$= \dfrac{{\left|\overset{\to}{b}\right|}^{2} - 2{\left|\overset{\to}{a}\right|}^{2} - \left(\overset{\to}{a} \cdot \overset{\to}{b}\right)}{\left|\overset{\to}{a} + \overset{\to}{b}\right|} = \dfrac{50 - 22 - \left(- 18\right)}{5} = \dfrac{46}{5}