Prove by vector method that $sinA/a=sinB/b=sinC/c$ ?

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Answer 1 · 2018-08-04T20:25:02

Please see below.

Let , $vec(AB)=bar(c) ,$ $vec(BC)=bar(a) ,$ $vec(CA)=bar(b)$

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So,

$bar(a)+bar(b)+bar(c)=bar(0)$

Using definition of cross Product

$bar(a)xx(bar(a)+bar(b)+bar(c))=bar(a)xxbar(0)$

$=>(bar(a)xxbar(a))+(bar(a) xxbar(b))+(bar(a) xxbar(c))=bar(0)to[becausebar(a)xxbar(0)=bar(0) ]$

$=>bar(0)+(bar(a) xxbar(b))+(bar(a) xxbar(c))=bar(0)to[because(bar(a)xxbar(a))=bar(0)]$

$=>(bar(a) xxbar(b))-(bar(c)xxbar(a))=bar(0)$

$=>(bar(a) xxbar(b))=(bar(c)xxbar(a))$

$=>a*bsin(pi-C)=c*asin(pi-B)$

$=>bsinC=csinB$

$=>sinC/c =sinB/b....to(1)$

Similarly we can prove that ,

$=>sinA/a =sinB/b....to(2)$

Hence ,

$sinA/a =sinB/b=sinC/c$

Prove by vector method that sinA/a=sinB/b=sinC/csinA/a=sinB/b=sinC/c?