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NDA 2010
Exam
NDA
Year
2010
Subject
Mathematics
Chapter
Vector Algebra

NDA 2010 · Mathematics · Vector Algebra

Consider the following statements: I. For any three vectors a , b ,...

Consider the following statements: I. For any three vectors a⃗,b⃗,c⃗\vec{a}, \vec{b}, \vec{c}; a⃗⋅[(b⃗+c⃗)×(a⃗+b⃗+c⃗)]=0\vec{a} \cdot[(\vec{b}+\vec{c}) \times(\vec{a}+\vec{b}+\vec{c})]=0 II. For any three coplanar unit vectors d⃗,e⃗,f⃗;(d⃗×e⃗)⋅f⃗=1\vec{d}, \vec{e}, \vec{f} ;(\vec{d} \times \vec{e}) \cdot \vec{f}=1 Which of the statements given above is/are correct?

Options

  1. AI only
  2. BII only
  3. CBoth I and II
  4. DNeither I nor II

Correct answer

C. Both I and II

Step-by-step solution

a⃗[b⃗×a⃗+b⃗×b⃗+b⃗×c⃗+c⃗×a⃗+c⃗×b⃗+c⃗×c⃗]=0a⃗[−c⃗+a⃗+b⃗−a⃗]=0or a⃗⋅b−a⃗⋅c⃗=0∴b⃗=c⃗\begin{aligned} & \vec{a}[\vec{b} \times \vec{a}+\vec{b} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}+\vec{c} \times \vec{b}+\vec{c} \times \vec{c}] \\ & =0 \\ & \vec{a}[-\vec{c}+\vec{a}+\vec{b}-\vec{a}]=0 \\ & \text{or } \vec{a} \cdot b-\vec{a} \cdot \vec{c}=0 \\ & \therefore \vec{b}=\vec{c}\end{aligned}$$
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