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NDA 2020 Maths · Mathematics · Vector Algebra · hard

If vec a,vec b,vec c are unit vectors such that vec a+vec b+vec c=vec 0, then vec a·vec b+vec b·vec c+vec c·vec a is equal to

Single correct option

If a⃗,b⃗,c⃗\vec a,\vec b,\vec c are unit vectors such that a⃗+b⃗+c⃗=0⃗\vec a+\vec b+\vec c=\vec 0, then a⃗⋅b⃗+b⃗⋅c⃗+c⃗⋅a⃗\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a is equal to
  1. A. −32-\dfrac32
  2. B. −3-3
  3. C. 32\dfrac32
  4. D. 00
Show answer and solution

Answer: (A) −32-\dfrac32

Square the given relation: ∣a⃗+b⃗+c⃗∣2=0=∣a⃗∣2+∣b⃗∣2+∣c⃗∣2+2(a⃗⋅b⃗+b⃗⋅c⃗+c⃗⋅a⃗)|\vec a+\vec b+\vec c|^2=0=|\vec a|^2+|\vec b|^2+|\vec c|^2+2(\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a). Since each vector is a unit vector, 0=3+2S0=3+2S, so S=−32S=-\dfrac32. The value −3-3 forgets the factor 22 in the expansion, and 00 would wrongly suggest the vectors are mutually perpendicular. The vectors in fact make 120∘120^\circ with each other.
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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