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Question
If `|veca`| = 3, `|vecb|` = 5, `|vecc|` = 4 and `veca + vecb + vecc` = `vec0`, then find the value of `(veca.vecb + vecb.vecc + vecc.veca)`.
Solution
Given, `|veca|` = 3, `|vecb|` = 5, `|vecc|` = 4
and `veca + vecb + vecc = vec0`
∴ `|veca + vecb + vecc| = |vec0|`
⇒ `|veca + vecb + vecc|^2` = 0
⇒ `(veca + vecb + vecc).(veca + vecb + vecc)` = 0
⇒ `veca.veca + veca.vecb + veca.vecc + vecb.veca + vecb.vecb + vecb.vecc + vecc.veca + vecc.vecb + vecc.vecc` = 0
⇒ `|veca|^2 + veca.vecb + vecc.veca + veca.vecb + |vecb|^2 + vecb.vecc + vecc.veca + vecb.vecc + |vecc|^2` = 0 ...`[∵ veca.vecb = vecb.veca, veca.vecc = vecc.veca, vecb.vecc = vecc.vecb]`
⇒ `|veca|^2 + |vecb|^2 + |vecc|^2 + 2(veca.vecb + vecb.vecc + vecc.veca)` = 0
⇒ `(3)^2 + (5)^2 + (4)^2 + 2(veca.vecb + vecb.vecc + vecc.veca)` = 0
⇒ `9 + 25 + 16 + 2(veca.vecb + vecb.vecc + vecc.veca)` = 0
⇒ `50 + 2(veca.vecb + vecb.vecc + vecc.veca)` = 0
⇒ `veca.vecb + vecb.vecc + vecc.veca` = –25
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