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 ?

Options

A. `(|vecc|^(2))/(2)-|veca|=12`
B. `(|vecc|^(2))/(2)-|veca|=30`
C. `|vecaxxvecb+veccxxveca|=48sqrt(3)`
D. `veca.vecb=-72`

Answer

A, C, D

Explanation

`a., c., d. veca+ vecb + vecc=0 `
` implies vecb+ vecc=-veca`
`implies |vecb|+ |vecc|^(2) +2cevb. vecc= |veca|^(2)`
`implies 48+ |vecc|^(2)+48=144`
`implies |vecc|^(2)=48`
`implies |vecc|= 4sqrt(3)`
`therefore ( |vecc|^(2))/(2) +|veca|=36`
futher ,
`veca+vecb=-vecc``implies |veca|^(2)+|vecb|+2veca.vecb=|vecc|^(2)`
` implies 144+48+2veca. vecb=48`
`:' veca+vecb+vecc=0`
`implies veca+vecb +c=0`
` therefore |vecaxxvecb+veccxxveca|`
`2 |vecaxxvecb|`
`= 2 sqrt(a^(2)b^(2)-(veca.vecb)^(2))`
`= 2sqrt((144)(48)-(72)^(2)= 48sqrt(3)`