Let $ \vec{a} = 2\hat{i} - 5\hat{j} + 5\hat{k} $ and $ \vec{b} = \hat{i} - \hat{j} + 3\hat{k} $. If $ \vec{c} $ is a vector such that

$ 2(\vec{a} \times \vec{c}) + 3(\vec{b} \times \vec{c}) = \vec{0} $

and $ (\vec{a} - \vec{b}) \cdot \vec{c} = -97 $, then

$ |\vec{c} \times \vec{k}|^2 $ is equal to

$ 2(\vec{a} \times \vec{c}) + 3(\vec{b} \times \vec{c}) = 0 $ $ \Rightarrow (2\vec{a} + 3\vec{b}) \times \vec{c} = 0 \Rightarrow \vec{c} \parallel (2\vec{a} + 3\vec{b}) $ $ \vec{c} = \lambda (2\vec{a} + 3\vec{b}) $ $ = \lambda (7\hat{i} - 13\hat{j} + 19\hat{k}) $ Now $ (\vec{a} - \vec{b}) \cdot \vec{c} = -97 $ $ \Rightarrow \lambda (7 + 52 + 38) = -97 $ $ \Rightarrow \lambda = -1 $ Now $ \vec{c} = -7\hat{i} + 13\hat{j} - 19\hat{k} $ $ \vec{c} \times \vec{k} = -7\hat{j} + 13\hat{i} $ $ |\vec{c} \times \vec{k}|^2 = 7^2 + 13^2 = 218 $