AI-তৈরি নমুনা উত্তর
প্রদত্ত ভেক্টর সমীকরণ:( a i ^ + b j ^ + k ^ ) × ( 2 i ^ + 2 j ^ + 3 k ^ ) = i ^ − j ^ \left(a\hat{i} + b\hat{j} + \hat{k}\right) \times \left(2\hat{i} + 2\hat{j} + 3\hat{k}\right) = \hat{i} - \hat{j} ( a i ^ + b j ^ + k ^ ) × ( 2 i ^ + 2 j ^ + 3 k ^ ) = i ^ − j ^
বামপক্ষ:( a i ^ + b j ^ + k ^ ) × ( 2 i ^ + 2 j ^ + 3 k ^ ) = ∣ i ^ j ^ k ^ a b 1 2 2 3 ∣ \left(a\hat{i} + b\hat{j} + \hat{k}\right) \times \left(2\hat{i} + 2\hat{j} + 3\hat{k}\right) = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a & b & 1 \\ 2 & 2 & 3 \end{vmatrix} ( a i ^ + b j ^ + k ^ ) × ( 2 i ^ + 2 j ^ + 3 k ^ ) = i ^ a 2 j ^ b 2 k ^ 1 3 = i ^ ( 3 b − 2 ) − j ^ ( 3 a − 2 ) + k ^ ( 2 a − 2 b ) = \hat{i}(3b - 2) - \hat{j}(3a - 2) + \hat{k}(2a - 2b) = i ^ ( 3 b − 2 ) − j ^ ( 3 a − 2 ) + k ^ ( 2 a − 2 b ) = ( 3 b − 2 ) i ^ + ( 2 − 3 a ) j ^ + 2 ( a − b ) k ^ = (3b - 2)\hat{i} + (2 - 3a)\hat{j} + 2(a - b)\hat{k} = ( 3 b − 2 ) i ^ + ( 2 − 3 a ) j ^ + 2 ( a − b ) k ^
প্রশ্নমতে,( 3 b − 2 ) i ^ + ( 2 − 3 a ) j ^ + 2 ( a − b ) k ^ = i ^ − j ^ + 0 k ^ (3b - 2)\hat{i} + (2 - 3a)\hat{j} + 2(a - b)\hat{k} = \hat{i} - \hat{j} + 0\hat{k} ( 3 b − 2 ) i ^ + ( 2 − 3 a ) j ^ + 2 ( a − b ) k ^ = i ^ − j ^ + 0 k ^
উভয়পক্ষের সহগ সমীকৃত করে পাই:3 b − 2 = 1 ⟹ 3 b = 3 ⟹ b = 1 3b - 2 = 1 \implies 3b = 3 \implies b = 1 3 b − 2 = 1 ⟹ 3 b = 3 ⟹ b = 1 2 − 3 a = − 1 ⟹ 3 a = 3 ⟹ a = 1 2 - 3a = -1 \implies 3a = 3 \implies a = 1 2 − 3 a = − 1 ⟹ 3 a = 3 ⟹ a = 1 2 ( a − b ) = 0 ⟹ a − b = 0 ⟹ a = b 2(a - b) = 0 \implies a - b = 0 \implies a = b 2 ( a − b ) = 0 ⟹ a − b = 0 ⟹ a = b a = 1 a = 1 a = 1 b = 1 b = 1 b = 1
অতএব, a = 1 a = 1 a = 1 b = 1 b = 1 b = 1