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BITSAT 2013
Exam
BITSAT
Year
2013
Subject
Mathematics
Chapter
Vector Algebra

BITSAT 2013 · Mathematics · Vector Algebra

If a , b , c are three unit vectors such that a + b + c = 0 , where...

If a⃗,b⃗,c⃗\vec{a}, \vec{b}, \vec{c} are three unit vectors such that a⃗+b⃗+c⃗=0→,\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}, where 0→\overrightarrow{0} is null vector, then a⃗⋅b⃗+b⃗⋅c⃗+c⃗⋅a⃗\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a} is :

Options

  1. A-3
  2. B-2
  3. C−32-\frac{3}{2}
  4. D0

Correct answer

C. −32-\frac{3}{2}

Step-by-step solution

We have a⃗+b⃗+c⃗=0→∣a⃗+b⃗+c⃗∣=0⇒∣a⃗+b⃗+c⃗∣2=0⇒∣a⃗∣2+∣b⃗∣2+∣c⃗∣2+2(a⃗⋅b⃗+b⃗⋅c⃗+c⃗⋅a⃗)=0⇒1+1+1+2(a⃗⋅b⃗+b⃗⋅c⃗+c⃗⋅a⃗)=0⇒a⃗⋅b⃗+bˉ⋅c⃗+cˉ⋅aˉ=−32\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}|\vec{a}+\vec{b}+\vec{c}|=0 \Rightarrow|\vec{a}+\vec{b}+\vec{c}|^{2}=0\Rightarrow|\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})=0\Rightarrow 1+1+1+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})=0\Rightarrow \vec{a} \cdot \vec{b}+\bar{b} \cdot \vec{c}+\bar{c} \cdot \bar{a}=-\frac{3}{2}
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