JEE MAIN - Physics (2021 - 25th February Morning Shift - No. 16)
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, $$\overrightarrow {AO} = 2\widehat i + 3\widehat j - 4\widehat k$$
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$$\overrightarrow {AB} + \overrightarrow {AC} + \overrightarrow {AD} + \overrightarrow {AE} + \overrightarrow {AF} + \overrightarrow {AG} + \overrightarrow {AH} $$,
if, $$\overrightarrow {AO} = 2\widehat i + 3\widehat j - 4\widehat k$$
$$ - 16\widehat i - 24\widehat j + 32\widehat k$$
$$16\widehat i + 24\widehat j - 32\widehat k$$
$$16\widehat i + 24\widehat j + 32\widehat k$$
$$16\widehat i - 24\widehat j + 32\widehat k$$
Explanation
We know,
$$ \because $$ $$\overrightarrow {OA} + \overrightarrow {OB} + \overrightarrow {OC} + \overrightarrow {OD} + \overrightarrow {OE} + \overrightarrow {OF} + \overrightarrow {OG} + \overrightarrow {OH} = \overrightarrow 0 $$
By triangle law of vector addition, we can write
$$\overrightarrow {AB} = \overrightarrow {AO} + \overrightarrow {OB} \,;\,\overrightarrow {AC} = \overrightarrow {AO} + \overrightarrow {OC} $$
$$\overrightarrow {AD} = \overrightarrow {AO} + \overrightarrow {OD} \,;\,\overrightarrow {AE} = \overrightarrow {AO} + \overrightarrow {OE} $$
$$\overrightarrow {AF} = \overrightarrow {AO} + \overrightarrow {OF} \,;\,\overrightarrow {AG} = \overrightarrow {AO} + \overrightarrow {OG} $$
$$\overrightarrow {AH} = \overrightarrow {AO} + \overrightarrow {OH} $$
Now
$$\overrightarrow {AB} + \overrightarrow {AC} + \overrightarrow {AD} + \overrightarrow {AE} + \overrightarrow {AF} + \overrightarrow {AG} + \overrightarrow {AH} $$
$$ = (7\overrightarrow {AO} ) + \overrightarrow {OB} + \overrightarrow {OC} + \overrightarrow {OD} + \overrightarrow {OE} + \overrightarrow {OF} + \overrightarrow {OG} + \overrightarrow {OH} $$
$$ = (7\overrightarrow {AO} ) + \overrightarrow 0 - \overrightarrow {OA} $$
$$ = (7\overrightarrow {AO} ) + \overrightarrow {AO} $$
$$ = 8\overrightarrow {AO} = 8(2\widehat i + 3\widehat j - 4\widehat k)$$
$$ = 16\widehat i + 24\widehat j - 32\widehat k$$
$$ \because $$ $$\overrightarrow {OA} + \overrightarrow {OB} + \overrightarrow {OC} + \overrightarrow {OD} + \overrightarrow {OE} + \overrightarrow {OF} + \overrightarrow {OG} + \overrightarrow {OH} = \overrightarrow 0 $$
By triangle law of vector addition, we can write
$$\overrightarrow {AB} = \overrightarrow {AO} + \overrightarrow {OB} \,;\,\overrightarrow {AC} = \overrightarrow {AO} + \overrightarrow {OC} $$
$$\overrightarrow {AD} = \overrightarrow {AO} + \overrightarrow {OD} \,;\,\overrightarrow {AE} = \overrightarrow {AO} + \overrightarrow {OE} $$
$$\overrightarrow {AF} = \overrightarrow {AO} + \overrightarrow {OF} \,;\,\overrightarrow {AG} = \overrightarrow {AO} + \overrightarrow {OG} $$
$$\overrightarrow {AH} = \overrightarrow {AO} + \overrightarrow {OH} $$
Now
$$\overrightarrow {AB} + \overrightarrow {AC} + \overrightarrow {AD} + \overrightarrow {AE} + \overrightarrow {AF} + \overrightarrow {AG} + \overrightarrow {AH} $$
$$ = (7\overrightarrow {AO} ) + \overrightarrow {OB} + \overrightarrow {OC} + \overrightarrow {OD} + \overrightarrow {OE} + \overrightarrow {OF} + \overrightarrow {OG} + \overrightarrow {OH} $$
$$ = (7\overrightarrow {AO} ) + \overrightarrow 0 - \overrightarrow {OA} $$
$$ = (7\overrightarrow {AO} ) + \overrightarrow {AO} $$
$$ = 8\overrightarrow {AO} = 8(2\widehat i + 3\widehat j - 4\widehat k)$$
$$ = 16\widehat i + 24\widehat j - 32\widehat k$$
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