InterviewSolution
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InterviewSolution
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Let `triangle PQR` be a triangle. Let ` veca = vec(QR) , vecb = vec(RP) and vecc= vec(PQ) . " if " |veca| = 12, |vecb| = 4sqrt3 and 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` |
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Answer» Correct Answer - 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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