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The length of the longer diagonal of the...

The length of the longer diagonal of the parallelogram constructed on ` 5veca + 2vecb and veca - 3vecb, ` if it is given that ` |veca|=2sqrt2, |vecb|=3 and veca. Vecb= pi/4` is

A

15

B

`sqrt3`

C

`sqrt593`

D

`sqrt369`

Text Solution

Verified by Experts

The correct Answer is:
C
The diagonals of the parallelogram are
` vecalpha = 5veca + 2vecb + veca -3vecb = 6veca - vecb and beta = +- ( 4 veca + 5vecb)`
Now,
` |vecalpha|= | 6 veca - vecb|`
` Rightarrow |vecalpha| = sqrt(36 |veca|^(2) + |vecb|^(2) -12 (veca .vecb))`
`|vecalpha|=sqrt(36xx8+9-12xx 2sqrt2xx3xx1/sqrt2) =15`
and,
` |vecbeta|=|4veca +5vecb|`
`|vecbeta|=sqrt(16|veca|^(2)+25|vecb|^(2)+40(veca.vecb))`
`|vecbeta|=sqrt(16xx8+25xx9 +40xx2sqrt2xx3xx1/sqrt2)=sqrt593`
Clearly, `|vecbeta| gt |vecalpha|`
Hence, the length of the longer diagonal is `sqrt593`
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