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NDA 2 2025 Maths, question 67

Topic: Vectors

Q67

How many of the following can be a vector perpendicular to both the vectors 2i^−j^+k^2\hat{i}-\hat{j}+\hat{k} and i^+j^+3k^\hat{i}+\hat{j}+3\hat{k}?

I. 4i^+5j^−3k^4\hat{i}+5\hat{j}-3\hat{k}

II. −8i^−10j^+6k^-8\hat{i}-10\hat{j}+6\hat{k}

III. 150(−4i^−5j^+3k^)\frac{1}{50}(-4\hat{i}-5\hat{j}+3\hat{k})

Select the correct answer.

  1. (a) None
  2. (b) One
  3. (c) Two
  4. (d) All threeCorrect answer

Solution.

Answer: (d) All three

  1. The cross product of (2,−1,1) and (1,1,3) is (−4,−5,3).
  2. All three displayed vectors are nonzero scalar multiples of it, so each has zero dot product with both original vectors.
  3. All three qualify.