Q If a→,b→,c→ are three vectors such that a→+b→+c→=0→, then prove that a→×b→=b→×c→=c→×a→, hence shpw that a→b→c→=0 - Maths - Vector Algebra

Question

Q   I f   a → , b → , c →   a r e
  t h r e e   v e c t o r s   s u c h   t h a t
  a → + b → + c → = 0 → ,   t h e
n   p r o v e   t h a t   a → × b
→ = b → × c → = c → × a →
,   h e n c e   s h p w   t h a t   a → b
→ c → = 0

Sanjari Kalantri · Accepted Answer

Dear Student,

Given: a→ + b→ + c→ = 0To prove: a →× b→ = b→ × c→ = c→ × a→; a→   b→   c→ = 0Proof: ∵ a→ + b→ + c→ = 0⇒ a→ + b→ = -c→⇒ a→ × a→ + b→ = a→ × -c→⇒ a→ × a→ + a→ × b→ = - a→ × c→⇒ a→ × b→ = c→ × a→                                     ∵ a→ × b→ = -b→ × a→, a→ × a→ = 0Similarly,we can show that a →× b→ = b→ × c→ and hence, a →× b→ = b→ × c→ = c→ × a→
Now, a→   b→   c→= a→
b→ × c→= a→
a→ × b→                                                  ∵ a →× b→ = b→ × c→ = c→ × a→ as proved above= a→   a→   b→= 0                                                                  ∵ a→   a→   b→ = a→   b→   a→ = b→   a→   a→ = 0Hence, proved

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Q . I f a → , b → , c → a r e t h r e e v e c t o r s s u c h t h a t a → + b → + c → = 0 → , t h e n p r o v e t h a t a → × b → = b → × c → = c → × a → , h e n c e s h p w t h a t a → b → c → = 0 .