If → a = ^ i + 2 ^ k , → b = ^ i + ^ j + ^ k , → c = 7 ^ i - 3 ^ j...

Given,

$\overset{\to}{r} \times \overset{\to}{b} + \overset{\to}{b} \times \overset{\to}{c} = \overset{\to}{0}$

$⇒ \overset{\to}{r} \times \overset{\to}{b} - \overset{\to}{c} \times \overset{\to}{b} = \overset{\to}{0}$

$⇒ \left(\overset{\to}{r} - \overset{\to}{c}\right) \times \overset{\to}{b} = 0$

$⇒ \left(\overset{\to}{r} - \overset{\to}{c}\right) | | \overset{\to}{b}$

Therefore, $\overset{\to}{r} - \overset{\to}{c} = \lambda\overset{\to}{b}$

$⇒ \overset{\to}{r} = \overset{\to}{c} + \lambda\overset{\to}{b}$

Also,

$\overset{\to}{r} \cdot \overset{\to}{a} = 0$ (given)

$⇒ \left(\overset{\to}{c} + \lambda\overset{\to}{b}\right) \cdot \overset{\to}{a} = 0$

$⇒ \overset{\to}{c} \cdot \overset{\to}{a} + \lambda\overset{\to}{b} \cdot \overset{\to}{a} = 0$

$⇒ \lambda = \dfrac{- \overset{\to}{c} \cdot \overset{\to}{a}}{\overset{\to}{b} \cdot \overset{\to}{a}}$

Now,

$\overset{\to}{r} \cdot \overset{\to}{c} = \left(\overset{\to}{c} + \lambda\overset{\to}{b}\right) \cdot \overset{\to}{c}$

$= \left(\overset{\to}{c} - \dfrac{\overset{\to}{c} \cdot \overset{\to}{a}}{\overset{\to}{b} \cdot \overset{\to}{a}}\overset{\to}{b}\right) \cdot \overset{\to}{c}$

$= {\left|\overset{\to}{c}\right|}^{2} - \left(\dfrac{\overset{\to}{c} \cdot \overset{\to}{a}}{\overset{\to}{b} \cdot \overset{\to}{a}}\right)\left(\overset{\to}{b} \cdot \overset{\to}{c}\right)$

$= 74 - \left(\dfrac{15}{3}\right) \times 8$

$= 74 - 40 = 34$