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Let vec(a),vec(b), & vec(c) be three vec...

Let `vec(a),vec(b), & vec(c)` be three vectors such that `|vec(b)| = 2|vec(a)| & |vec(c)| = 3|vec(a)|`. The Angle between each pair of vectors is `60^(@)` such that `|vec(a) + 2vec(b) + 3vec(c)| = sqrt(21)` then `sqrt(7)| vec(c)|` is equal to

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Verified by Experts

The correct Answer is:
`3.00`
`|vec(a)| = k`
`|vec(b)| = 2k`
`|vec(c)| = 3k`
`|vec(a) + 2vec(b) + 3vec(c)| = sqrt(21)`
`(|vec(a) + 2vec(b) + 3vec(c)|)^(2) = 21`
`a^(2) + 4b^(2) + 9c^(2) + 2[2vec(a) cdot vec(b) + 6vec(b) cdot vec(c) + 3vec(c) cdot vec(a)] = 21`
`k^(2) + 16k^(2) + 81k^(2) + 2[(4k^2)/2 + (36k^2)/2 + (9k^2)/2] = 21`
`147k^(2) = 21 implies k = +- 1/(sqrt7)`
`|c| = 3k = 3/(sqrt7)` [magnitude is always +ve]`
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