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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

`1/2|vecc|^(2) -|veca| =12`

B

`1/2|vecc|^(2) + |veca| =30`

C

` |veca xx vecb + vecc xx veca| = 48sqrt3`

D

`veca.vecb= -72`

Text Solution

Verified by Experts

The correct Answer is:
A, C, D
`veca= vec(QR) ,vecb = vec(RP) and vecc= vec(PQ)`
` Rightarrow veca + vecb + vecc = vec(QR) + vec(RP) + vec(PQ)`
` Rightarrow veca + vecb + vecc = vec(OQ) `
` Rightarrow veca + vecb + vecc= vec0`
` Rightarrow vecb + vecc = -veca`
` Rightarrow vecb + vecc = |-veca|`
` |vecb + vecc|^(2) =|veca|^(2)`
` |vecb|^(2) +|vecc|^(2) + 2(vecb .vecc) = |veca|^(2) `
` 48 + |vecc|^(2) + 48 = 144 `
` |vecc| = 4sqrt3`
` 1/2 |vecc|^(2) - |veca| = 24 -12 =12 and 1/2 |vecc|^(2) + | veca| = 24 + 12 =36`
So, option (a) is true and option (b) is not true. Again.
` veca + vecb + vecc= vec0`
` Rightarrow |veca + vecb| = | -vecc|`
` Rightarrow |veca + vecb|^(2) +2 (veca.vecb) = |vecc|^(2)`
` Rightarrow 144 + 48+2 (veca .vecb) = 48`
` Rightarrow veca.vecb = -72`
So, option (d) is true.
Again
` veca + vecb + vecc= vec0`
` Rightarrow veca xx ( veca + vecb + vecc) = veca xx vec0`
` Rightarrow veca xx veca + veca xx vecb + veca xx vecc= vec0`
` Rightarrow veca xx vecb = vecc xx veca `
` therefore |veca xx vecb + vecc xx veca| = |2 (veca xx vecb)| = 2|veca xx vecb|`
` 2 sqrt(|veca|^(2) |vecb|^(2) -(veca -vecb)^(2) )`
` 2 sqrt(144xx 48 -(-72) ^(2)) = 48sqrt3`
So, option (c) is correct
Hence, option (a),(c) and (d) are ture.
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