In an octagon ABCDEFGH of equal side, what is the sum of AB + AC + AD + AE + AF + AG + AH, if, AO = 2i + 3j - 4k (a) -16i - 24j + 32k (b) 16i + 24j - 32k (c) 16i + 24j + 32k (d) 16i - 24j + 32k

Answer: (b). AX = AO + OX for each of the 7 vertices B to H. Sum = 7AO + (OB + + OH). For a regular octagon all 8 vectors from the centre add to zero, so OB + + OH = -OA = AO. Sum = 8AO = 16i + 24j - 32k, answer (b).

In an octagon ABCDEFGH of equal side, what is the sum of AB + AC + AD… | JEE Main 2021 (25 Feb, Shift 1)

Q23JEE Main 2021 · 25 FEB · SHIFT 1Difficult

Addition & Subtraction of Vectors

In an octagon ABCDEFGH of equal side, what is the sum of

$\overrightarrow{AB} + \overrightarrow{AC} + \overrightarrow{AD} + \overrightarrow{AE} + \overrightarrow{AF} + \overrightarrow{AG} + \overrightarrow{AH}$ ,

if, $\overrightarrow{AO} = 2\hat{i} + 3\hat{j} - 4\hat{k}$

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Answer: (b)

$\overrightarrow{AX} = \overrightarrow{AO} + \overrightarrow{OX}$ for each of the 7 vertices B to H.
Sum $= 7\overrightarrow{AO} + (\overrightarrow{OB} + \dots + \overrightarrow{OH})$ .
For a regular octagon all 8 vectors from the centre add to zero, so $\overrightarrow{OB} + \dots + \overrightarrow{OH} = -\overrightarrow{OA} = \overrightarrow{AO}$ .
Sum $= 8\overrightarrow{AO} = 16\hat{i} + 24\hat{j} - 32\hat{k}$ , answer (b).

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