Let \( \vec{a} = 3\hat{i} + 2\hat{j} + \hat{k} \), \( \vec{b} = 2\hat{i} - \hat{j} + 3\hat{k} \), and \( \vec{c} \) be a vector such that \((\vec{a} + \vec{b}) \times \vec{c} = 2(\vec{a} \times \vec{b}) + 24\hat{j} - 6\hat{k}\) and \((\vec{a} - \vec{b} + \hat{i}) \cdot \vec{c} = -3.\) Then \( |\vec{c}|^2 \) is equal to \(\_\_\_\_\_.\)

Let a ⃗ = 3 i ^ + 2 j ^ + k ^ a =3 i ^ +2 j ^ + k ^ , b ⃗ = 2 i ^ − j ^ + 3 k ^ b =2 i ^ − j ^ +3 k ^ , and c ⃗ c be a vector such that ( a ⃗ + b ⃗ ) × c ⃗ = 2 ( a ⃗ × b ⃗ ) + 24 j ^ − 6 k ^ ( a + b )× c =2( a × b )+24 j ^ −6 k ^ and ( a ⃗ − b ⃗ + i ^ ) ⋅ c ⃗ = − 3. ( a − b + i ^ )⋅ c =−3.Then ∣ c ⃗ ∣ 2 ∣ c ∣ 2 is equal to _ _ _ _ _ . ____

List-I	List-II
(A) 4î − 2ĵ − 4k̂	(I) A vector perpendicular to both î + 2ĵ + k̂ and 2î + 2ĵ + 3k̂
(B) 4î − 4ĵ + 2k̂	(II) Direction ratios are −2, 1, 2
(C) 2î − 4ĵ + 4k̂	(III) Angle with the vector î − 2ĵ − k̂ is cos⁻¹(1/√6)
(D) 4î − ĵ − 2k̂	(IV) Dot product with −2î + ĵ + 3k̂ is 10