If the projection of the vector ^ i + 2 ^ j + ^ k on the sum of the...

Sum of two vectors = 2\overset{^}{i} + 4\overset{^}{j} - 5\overset{^}{k} + ( - \lambda\overset{^}{i} + 2\overset{^}{j} + 3\overset{^}{k} ) = ( 2 - \lambda ) \overset{^}{i} + 6\overset{^}{j} - 2\overset{^}{k}
Projection $= 1 = \dfrac{1 ( 2 - \lambda ) + 2 ( 6 ) + 1 \times ( - 2 )}{\sqrt{( 2 - \lambda{)}^{2} + {6}^{2} + {2}^{2}}}$

$⇒ \dfrac{12 - \lambda}{\sqrt{{\lambda}^{2} - 4\lambda + 44}} = 1$
$⇒ ( 12 - \lambda{)}^{2} = {\lambda}^{2} - 4\lambda + 44$
$⇒ 144 + {\lambda}^{2} - 24\lambda = {\lambda}^{2} - 4\lambda + 44$
$⇒ 100 = 20\lambda$
$⇒ \lambda = 5$