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If vec c = 3vec a+4vec b and 2vec c =vec...

If `vec c = 3vec a+4vec b and 2vec c =vec a -3vec b`, show that (i) `vec c and vec a` have the same direction and `|vec c| gt |vec a|`(ii) `vec b and vec c` have opposite direction and `|vec c| gt |vec b|`

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We have,
`c=3a+4b and 2c=a-3b`
`implies2(3a+4b)=a-3b`
`implies5a=-11b`
`implies a=-(11)/(5)b and b=-(5)/(11)a`
(i) `c=3a+b=3a+4(-(5)/(11)a)" "(using" "b=-(5)/(11)a)`
`=3a-(20)/(11)a=(13)/(11)a`
which shows that c and a have the same direction.
and `c=(13)/(11)a `
`implies|c|=(13)/(11)|a|implies|c|gt|a|`
(ii) we have, `c=3a+4b and a=-(11)/(5)b`
`c=3(-(11)/(5)b)+4b=-(33)/(5)b+4b`
`c=-(13)/(5)b`
this shows c and b have opposite directions. ,brgt also `|c|=|-(13)/(5)b|=(13)/(5)|b|implies|c|gt|b|`.
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