Let → a = 3 ^ i + ^ j - 2 ^ k , → b = 4 ^ i + ^ j + 7 ^ k and → c =...

Given: $\overset{\to}{p} \times \overset{\to}{b} = \overset{\to}{c} \times \overset{\to}{b}$

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

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

$⇒ \left(\overset{\to}{p} - \overset{\to}{c}\right) = k\overset{\to}{b}$

$⇒ \overset{\to}{p} = k\overset{\to}{b} + \overset{\to}{c}$

⇒ \overset{\to}{p} = \left(4k + 1\right)\overset{^}{i} + \left(k - 3\right)\overset{^}{j} + \left(7k + 4\right)\overset{^}{k}

Now, $\overset{\to}{p} . \overset{\to}{a} = 0$

$⇒ 3\left(4k + 1\right) + \left(k - 3\right) - 2\left(7k + 4\right) = 0$

$⇒ k = - 8$

⇒ \overset{\to}{p} = - 31\overset{^}{i} - 11\overset{^}{j} - 52\overset{^}{k}

⇒ \overset{\to}{p} . \left(\overset{^}{i} - \overset{^}{j} - \overset{^}{k}\right) = - 31 + 11 + 52

⇒ \overset{\to}{p} . \left(\overset{^}{i} - \overset{^}{j} - \overset{^}{k}\right) = 32