Let → a = ^ i + 2 ^ j + λ ^ k , → b = 3 ^ i - 5 ^ j - λ ^ k , → a ·...

Given:

$\overset{\to}{a} \times \overset{\to}{c} = \overset{\to}{b} \times \overset{\to}{c}$

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

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

So, \left(\overset{\to}{a} - \overset{\to}{b}\right) ∥ \overset{\to}{c}$, hence</p><p>$\overset{\to}{a} - \overset{\to}{b} = \mu\overset{\to}{c}$</p><p>where,$\mu$is any scalar.</p><p>$⇒ - 2\overset{^}{i} + 7\overset{^}{j} + 2\lambda\overset{^}{k} = \mu\overset{\to}{c} . . . . \left(1\right)$</p><p>Taking dot product with$\overset{\to}{a}$, we get</p><p>$⇒ - 2 + 14 + 2{\lambda}^{2} = 7\mu$</p><p>$⇒ 7\mu = 2{\lambda}^{2} + 12 . . . \left(2\right)

Taking dot product$with \overset{\to}{b} in$\left(1\right)$, we get</p><p>$- 6 - 35 - 2{\lambda}^{2} = - \dfrac{43}{2}\mu$</p><p>$⇒ 82 + 4{\lambda}^{2} = 43\mu . . . . \left(3\right)

Solving $\left(2\right)\text{\text{ and } }$\left(3\right), we get

${\lambda}^{2} = 1, \mu = 2$

So,

$\overset{\to}{a} \cdot \overset{\to}{b} = 3 - 10 - {\lambda}^{2} = - 8$

$⇒ \left|\overset{\to}{a} \cdot \overset{\to}{b}\right|$ = 8$</p>$