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NDA 2019 Maths · Mathematics · Vector Algebra · medium

Three vectors are coplanar if and only if their scalar triple product is zero. 2. Interchanging any two of the three vectors in [vec a vec b vec c]…

Single correct option

Consider the following statements:
1. Three vectors are coplanar if and only if their scalar triple product is zero.
2. Interchanging any two of the three vectors in [a⃗  b⃗  c⃗][\vec a\;\vec b\;\vec c] changes its sign.
Which of the statements given above is/are correct?
  1. A. 1 only
  2. B. 2 only
  3. C. Both 1 and 2
  4. D. Neither 1 nor 2
Show answer and solution

Answer: (C) Both 1 and 2

Statement 1: the scalar triple product equals the volume of the parallelepiped (up to sign), which vanishes exactly when the three vectors lie in one plane; true. Statement 2: swapping two rows of the determinant representing [a⃗  b⃗  c⃗][\vec a\;\vec b\;\vec c] reverses its sign, e.g. [b⃗  a⃗  c⃗]=−[a⃗  b⃗  c⃗][\vec b\;\vec a\;\vec c]=-[\vec a\;\vec b\;\vec c]; true. Both are correct.
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Original question written in the NDA 2019 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake