In an octagon ABCDEFGH of equal sides- what is the sum of AB-AC-AD-AE-AF-AG-AH if AO-2-3-4k- -

From figure, nbsp;AO+OB=AB AO+OC=AC AO+OD=AD AO+OE=AE AO+OF=AF AO+OG=AG AO+OH=AH On adding all the above equations, we get, nbsp; 8AO=AB+AC+AD+AE+AF+AG+A ...

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Byju's Answer

Standard IX

Mathematics

Condition for Three Vectors to Form a Triangle

In an octagon...

Question

In an octagon $A B C D E F G H$ of equal sides, what is the sum of $- - \to A B + - - \to A C + - - \to A D + - - \to A E + - - \to A F + - - \to A G + - - \to A H$ if $- - \to A O = 2^i + 3^j - 4^k ?$

16^i + 24^j - 32^k

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- 16^i - 24^j + 32^k

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- 16^i + 24^j + 32^k

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Solution

Step 1: Given that:
An octagon ABCDEFGH with centre O.
In which, AB= BC= CD = DE= EF= FG= GH= HA

$\to A O = 2^i + 3^j - 4^k$
Step 2: Formula used:
According to the triangle law of vector addition;
If two vectors $\to P$ and $\to Q$ are represented by the sides AB and BC of as a triangle in both magnitude and direction then the resultant of both the vectors is represented in magnitude and direction by the third side of the triangle taken in the opposite order that is by AC.

Step 2: Calculation of $\to A B + \to A C + \to A D + \to A E + \to A F + \to A G + \to A H$
Joining all the vertices of the octagon with point O, we have;

All the sides, in each part of the octagon, are represented by a vector.
Therefore, using triangle law of vector addition, we get;
$\to A O + \to O B = \to A B$ .........(1)
$\to A O + \to O C = \to A C$ ..........(2)
$\to A O + \to O D = \to A D$ .........(3)
$\to A O + \to O E = \to A E$ ..........(4)
$\to A O + \to O F = \to A F$ .........(5)
$\to A O + \to O G = \to A G$ .........(6)
$\to A O + \to O H = \to A H$ .........(7)
Now,
$\to O A = - \to \to O E a n d \to \to A O = \to \to O E$
$\to O B = - \to O F$
$\to O C = - \to O G$
$\to O D = - \to O H$
Using these values and adding equations (1). (2), (3), (4), (5), (6) and (7), we get
$7 \to A O + \to A O = \to A B + \to A C + \to A D + \to A E + \to A F + \to A G + \to A H$
$8 \to A O = \to A B + \to A C + \to A D + \to A E + \to A F + \to A G + \to A H$ ...........(8)
Putting the value of $\to A O$ , in equation (8) we get;
$\to A B + \to A C + \to A D + \to A E + \to A F + \to A G + \to A H = 8 (2^i + 3^j - 4^k)$
$\to A B + \to A C + \to A D + \to A E + \to A F + \to A G + \to A H = 16^i + 24^j - 32^k$
Thus,
The value of $\to A B + \to A C + \to A D + \to A E + \to A F + \to A G + \to A H = 16^i + 24^j - 32^k$

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Similar questions

Q. In an octagon

A B C D E F G H

of equal sides, what is the sum of

- - \to A B + - - \to A C + - - \to A D + - - \to A E + - - \to A F + - - \to A G + - - \to A H

- - \to A O = 2^i + 3^j - 4^k ?

Q. If

2^i

and

^j + \sqrt{3}^j

are the vectors representing sides

A B

and

B C

respectively of a regular hexagon

A B C D E F

. Express vectors

¯ ¯¯¯¯¯¯¯ ¯ C D, ¯ ¯¯¯¯¯¯¯ ¯ D E, ¯ ¯¯¯¯¯¯ ¯ F A, ¯ ¯¯¯¯¯¯ ¯ A C, ¯ ¯¯¯¯¯¯¯ ¯ A D, ¯ ¯¯¯¯¯¯ ¯ A E, ¯ ¯¯¯¯¯¯ ¯ C E

Q. If ABCDEF be a regular hexagon with centre 'O' then show that

¯ ¯¯¯¯¯¯ ¯ A B + ¯ ¯¯¯¯¯¯ ¯ A C + ¯ ¯¯¯¯¯¯¯ ¯ A D + ¯ ¯¯¯¯¯¯ ¯ A E + ¯ ¯¯¯¯¯¯ ¯ A F = 3 ¯ ¯¯¯¯¯¯¯ ¯ A D = 6 ¯ ¯¯¯¯¯¯ ¯ A O

Q. In a regular hexagon

A B C D E F

¯ ¯¯¯¯¯¯ ¯ A B + ¯ ¯¯¯¯¯¯ ¯ A C + ¯ ¯¯¯¯¯¯¯ ¯ A D + ¯ ¯¯¯¯¯¯ ¯ A E + ¯ ¯¯¯¯¯¯ ¯ A F = k ¯ ¯¯¯¯¯¯¯ ¯ A D

, then

k

is equal to

Q. If

A B C D E F

is a regular hexagon, show that

¯ ¯¯¯¯¯¯ ¯ A B + ¯ ¯¯¯¯¯¯ ¯ A C + ¯ ¯¯¯¯¯¯¯ ¯ A D + ¯ ¯¯¯¯¯¯ ¯ A E + ¯ ¯¯¯¯¯¯ ¯ A F = 6 ¯ ¯¯¯¯¯¯ ¯ A O

, where

O

is the centre of the hexagon.

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In an octagon ABCDEFGH of equal sides, what is the sum of −−→AB+−−→AC+−−→AD+−−→AE+−−→AF+−−→AG+−−→AH if −−→AO=2^i+3^j−4^k ?

In an octagon $A B C D E F G H$ of equal sides, what is the sum of $- - \to A B + - - \to A C + - - \to A D + - - \to A E + - - \to A F + - - \to A G + - - \to A H$ if $- - \to A O = 2^i + 3^j - 4^k ?$