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NDA 1 2025 Maths, question 67

Topic: Vectors

Q67

Consider the following statements in respect of a vector d→=(a→×b→)×c→\overrightarrow{d} = \left( \overrightarrow{a} \times \overrightarrow{b} \right) \times \overrightarrow{c}: I. d→\overrightarrow{d} is coplanar with a→\overrightarrow{a} and b→\overrightarrow{b}. II. d→\overrightarrow{d} is perpendicular to c→\overrightarrow{c}. Which of the statements given above is/are correct?

  1. (a) I only
  2. (b) II only
  3. (c) Both I and IICorrect answer
  4. (d) Neither I nor II

Solution.

Answer: (c) Both I and II

  1. The vector triple-product identity gives d→=b→(a→⋅c→)−a→(b→⋅c→)\overrightarrow{d} = \overrightarrow{b}\left( \overrightarrow{a} \cdot \overrightarrow{c} \right) - \overrightarrow{a}\left( \overrightarrow{b} \cdot \overrightarrow{c} \right).
  2. Thus d→\overrightarrow{d} is a linear combination of a→\overrightarrow{a} and b→\overrightarrow{b}, proving I.
  3. Also, any cross product with c→\overrightarrow{c} has zero dot product with c→\overrightarrow{c}: d→⋅c→=0\overrightarrow{d} \cdot \overrightarrow{c} = 0.
  4. Thus II is true in the standard vector sense of orthogonality.