For any vector a , the value of |a × i|^2 + |a × j|^2 + |a × k|^2 is equal to: - Mathematics MCQ AIEEE 2026 | ExamTest.live

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For any vector a, the value of ∣ a \times \hat{i} ∣^{2} + ∣ a \times \hat{j} ∣^{2} + ∣ a \times \hat{k} ∣^{2} is equal to:

Solution & Step-by-step Explanation

Let a = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k} .Then a \times \hat{i} = (a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}) \times \hat{i} = - a_{2} \hat{k} + a_{3} \hat{j} . ∣ a \times \hat{i} ∣^{2} = a_{2}^{2} + a_{3}^{2} .Similarly: ∣ a \times \hat{j} ∣^{2} = a_{1}^{2} + a_{3}^{2} ∣ a \times \hat{k} ∣^{2} = a_{1}^{2} + a_{2}^{2} Adding them: ∣ a \times \hat{i} ∣^{2} + ∣ a \times \hat{j} ∣^{2} + ∣ a \times \hat{k} ∣^{2} = (a_{2}^{2} + a_{3}^{2}) + (a_{1}^{2} + a_{3}^{2}) + (a_{1}^{2} + a_{2}^{2}) = 2 (a_{1}^{2} + a_{2}^{2} + a_{3}^{2}) = 2∣ a ∣^{2} .

Try it yourself before checking the explanation above.

For any vector a, the value of ∣ a \times \hat{i} ∣^{2} + ∣ a \times \hat{j} ∣^{2} + ∣ a \times \hat{k} ∣^{2} is equal to:

3∣ a ∣^{2}

∣ a ∣^{2}

2∣ a ∣^{2}

4∣ a ∣^{2}

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