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JEE PYQ: Vector Algebra - Question ID 544540e3dbbc (JEE Main 2021)

ID: 544540e3dbbcJEE Main 2021Single Correct MCQ
In an octagon ABCDEFGH of equal side, what is the sum of

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

if, AO→=2i^+3j^−4k^\overrightarrow {AO} = 2\widehat i + 3\widehat j - 4\widehat k

JEE Main 2021 (Online) 25th February Morning Shift Physics - Vector Algebra Question 33 English
JEE Question illustration 544540e3dbbc

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Step-by-step Explanation

We know,

∵\because OA→+OB→+OC→+OD→+OE→+OF→+OG→+OH→=0→\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

AB→=AO→+OB→ ; AC→=AO→+OC→\overrightarrow {AB} = \overrightarrow {AO} + \overrightarrow {OB} \,;\,\overrightarrow {AC} = \overrightarrow {AO} + \overrightarrow {OC}

AD→=AO→+OD→ ; AE→=AO→+OE→\overrightarrow {AD} = \overrightarrow {AO} + \overrightarrow {OD} \,;\,\overrightarrow {AE} = \overrightarrow {AO} + \overrightarrow {OE}

AF→=AO→+OF→ ; AG→=AO→+OG→\overrightarrow {AF} = \overrightarrow {AO} + \overrightarrow {OF} \,;\,\overrightarrow {AG} = \overrightarrow {AO} + \overrightarrow {OG}

AH→=AO→+OH→\overrightarrow {AH} = \overrightarrow {AO} + \overrightarrow {OH}

Now

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

=(7AO→)+OB→+OC→+OD→+OE→+OF→+OG→+OH→= (7\overrightarrow {AO} ) + \overrightarrow {OB} + \overrightarrow {OC} + \overrightarrow {OD} + \overrightarrow {OE} + \overrightarrow {OF} + \overrightarrow {OG} + \overrightarrow {OH}

=(7AO→)+0→−OA→= (7\overrightarrow {AO} ) + \overrightarrow 0 - \overrightarrow {OA}

=(7AO→)+AO→= (7\overrightarrow {AO} ) + \overrightarrow {AO}

=8AO→=8(2i^+3j^−4k^)= 8\overrightarrow {AO} = 8(2\widehat i + 3\widehat j - 4\widehat k)

=16i^+24j^−32k^= 16\widehat i + 24\widehat j - 32\widehat k
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