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Question
Find `|veca| and |vecb|`, if `(veca + vecb).(veca -vecb) = 8 and |veca| = 8|vecb|.`
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Solution
We have, `(veca + vecb) xx (veca - vecb) = 8`
⇒ `veca xx veca - veca xx vecb + vecb xx veca - vecb xx vecb = 8`
but, `veca xx vecb = vecb xx veca`
∴ `veca xx veca - veca xx vecb + veca xx vecb - vecb xx vecb = 8`
= `veca xx veca . vecb xx vecb = 8`
= `64|vecb|^2 - |vecb|^2 = 8` `[∵ |veca| = 8|vecb|]`
= `63|vecb|^2 = 8`
∴ `|vecb| = sqrt(8/63) = 2/3sqrt(2/7)`
But `|veca| = 8 |vecb|`
⇒ `|veca| = (8sqrt8)/sqrt63`
`= (16sqrt2)/(3sqrt7)`
Hence, `|veca| = (16sqrt2)/(3sqrt7)`
and `|vecb| = (2sqrt2)/(3sqrt7)`
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