Let `Delta PQR` be a triangle Let `veca=bar(QR),vecb=bar(RP)and vecc=bar(PQ)if |veca|=12,|vecb|=4 sqrt(3)and vecb.vecc=24,`then which of the following is (are ) true ?

Options

A. `|vecc|^(2)/2-|veca|=12`
B. `|vecc|^(2)/2-|veca|=30`
C. `|vecaxxvecb + veccxxveca|= 48sqrt3`
D. `veca.vecb=-72`

Answer

a,c,d

Explanation

`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`