If $\vec{a} = \hat{i} - \hat{j} + 7\hat{k}$ and $\vec{b} = 5\hat{i} - \hat{j} + \lambda\hat{k}$, then find the value of $\lambda$ so that vectors $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$ are orthogonal.

$\pm 5$

The correct answer is Option (2) → $\pm 5$ ##

$\vec{a} = \hat{i} - \hat{j} + 7\hat{k}$ and $\vec{b} = 5\hat{i} - \hat{j} + \lambda\hat{k}$

Hence,

$\vec{a} + \vec{b} = 6\hat{i} - 2\hat{j} + (7 + \lambda)\hat{k}$ and $\vec{a} - \vec{b} = -4\hat{i} + (7 - \lambda)\hat{k}$

$\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$ will orthogonal if, 

$(\vec{a} + \vec{b})\cdot (\vec{a} - \vec{b})=0$

i.e., if $-24+(49 - \lambda^2) = 0$

$⇒\lambda^2 = 25$

$⇒\lambda = \pm 5$