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Let PQR be a triangle . Let veca=overlin...

Let `PQR` be a triangle . Let `veca=overline(QR),vecb = overline(RP) and vecc= overline(PQ).if |veca|=12, |vecb|=4sqrt3and vecb.vecc= 24` then which of the following is (are) true ?

A

`|vecc|^(2)/2-|veca|=12`

B

`|vecc|^(2)/2-|veca|=30`

C

`|vecaxxvecb + veccxxveca|= 48sqrt3`

D

`veca.vecb=-72`

Text Solution

Verified by Experts

The correct Answer is:
a,c,d
`veca + vecb+vecc =0`
`Rightarrow vecb + vecc= -veca`
`Rightarrow |vecb|^(2) +|vecc|^(2) + 2vecb.vecc= |veca|^(2)`
` Rightarrow 48 + |vec|^(2) + 48 = 144 `
` Rightarrow |vecc|^(2)=48`
`|vecc|= 4sqrt3`
` (|vecc|)^(2))/2+|veca|=36`
Further,
`veca+vecb=-vecc`
`Rightarrow |veca|^(2)+|vecb|^(2)+2veca.vecb=|vecc|^(2)`
`Rightarrow 144 + 48 + 2 veca. vecb= 48`
`veca. vecb = -72`
`veca.vecb + vecc=0`
`Rightarrow veca xx vecb +veca xx vecc=0`
`|veca xx vecb +vecc xx veca|`
`2|veca xx vecb|`
`=2 sqrt(a^(2)b^(2)-(veca.vecb)^(2))`
`2sqrt((144)(48)-(72)^(2))=48sqrt3`
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