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NDA 2020 Maths · Mathematics · Vector Algebra · easy

Two statements about the magnitudes of vector sums and products are given below, for any vectors vec a and vec b. 1. |vec a+vec b|≤|vec a|+|vec b|.…

Single correct option

Two statements about the magnitudes of vector sums and products are given below, for any vectors a⃗\vec a and b⃗\vec b.
1. ∣a⃗+b⃗∣≤∣a⃗∣+∣b⃗∣|\vec a+\vec b|\le|\vec a|+|\vec b|.
2. ∣a⃗×b⃗∣=∣a⃗∣∣b⃗∣cos⁡θ|\vec a\times\vec b|=|\vec a||\vec b|\cos\theta, where θ\theta is the angle between them.
Which of the statements is/are correct?
  1. A. 1 only
  2. B. 2 only
  3. C. Both 1 and 2
  4. D. Neither 1 nor 2
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Answer: (A) 1 only

Statement 1 is the triangle inequality: the length of one side of a triangle cannot exceed the sum of the other two, with equality only for vectors in the same direction. It is correct. The magnitude of the cross product is ∣a⃗∣∣b⃗∣sin⁡θ|\vec a||\vec b|\sin\theta, not cos⁡θ\cos\theta, so statement 2 is false; the cosine belongs to the dot product. For perpendicular vectors the cross product has its greatest magnitude, while cos⁡θ=0\cos\theta=0.
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Original question written in the NDA 2020 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