Let → a = 2 ^ i - 7 ^ j + 5 ^ k , → b = ^ i + ^ k and → c = ^ i + 2...

Given data as below:

\overset{\to}{a} = 2\overset{^}{i} - 7\overset{^}{j} + 5\overset{^}{k}

\overset{\to}{b} = \overset{^}{i} + \overset{^}{k}

\overset{\to}{c} = \overset{^}{i} + 2\overset{^}{j} - 3\overset{^}{k}

Given, $\overset{\to}{r} \times \overset{\to}{a} = \overset{\to}{c} \times \overset{\to}{a}$

$⇒ ( \overset{\to}{r} - \overset{\to}{c} ) \times \overset{\to}{a} = 0$

It means $( \overset{\to}{r} - \overset{\to}{c} )$ is parallel to $\overset{\to}{a}$,

$∴ \overset{\to}{r} = \overset{\to}{c} + \lambda\overset{\to}{a} . . . . . . . ( 1 )$

Given, $\overset{\to}{r} \cdot \overset{\to}{b} = 0$

From equation (1),

\left(\overset{\to}{c} + \lambda\overset{\to}{a}\right) \cdot \overset{\to}{b} = 0$</p><p>$⇒ \left(\overset{\to}{c} \cdot \overset{\to}{b}\right) + \lambda\left(\overset{\to}{b} \cdot \overset{\to}{a}\right) = 0$</p><p>$⇒ - 2 + \lambda ( 7 ) = 0 ⇒ \lambda = \dfrac{2}{7}$</p><p>On putting values in equation (1) we get,</p><p>$∴ \overset{\to}{r} = \overset{\to}{c} + \dfrac{2}{7}\overset{\to}{a} = \dfrac{1}{7} ( 11\overset{^}{i} - 11\overset{^}{k} )$</p><p>$| \overset{\to}{r} | = \dfrac{11\sqrt{2}}{7}$