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. `(|c|^(2))/(2)-|a|=12`
B. `(|c|^(2))/(2)+|a|=30`
C. `|axxb+cxxa|=48sqrt(3)`
D. `a.b =-72`

Answer

A, C, D

Explanation

Given `|a| =12 , |b| =4sqrt(3)`
`a+b+c=0`
`rArr a=- (b+c)`
We have `|a|^(2) =|b+c|^(2)`
`rArr |a|^(2) =|b|^(2) +|c|^(2)+2b.c`
`rAr 144 = 48 +|c|^(2) +48`

rArr |c| =4 sqrt(3) `
`Also , ` |c|^(2) =|a|^(2)+ |b|^(2) +2a.b`
` rArr 48 =144 + 48 + 2a. b`
`rArr a. b =- 72`
`:. ` Option (d) is correct.
Also , `axx b= c xx a`
`rArr a xx b + c xx a = 2a xx b `
`rArr | axx b+ c xx a| = 2 | a xx b| = 2 sqrt(|a|^(2) |b|^(2) -(a.b)^(2))`
`= 2 sqrt((144) (48) -(-72)^(2))`
`=2 (12) sqrt(48-36) =48 sqrt(3)`
`:. ` Oprion (c ) is correct .
Also `(|c|^(2))/(2) - |a| =24 -12 =12 `
`:.` Option (a) is correct.
and `(|c|^(2))/(2) + |a| =24 +12 =36`
`:. ` Option (b) is not correct .