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Consider A, B, C or D with position vect...

Consider A, B, C or D with position vectors : `7 hati - 4 hatj + 7 hatk, hati - 6 hatj + 10 hatk, - hati - 3 hat + 4 hatk and 5 hati - hatj + 5 hatk` respectively. Then ABCD is a:

A

square

B

rhombus

C

rectangle

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
D
Given that, `OA=7hati-4hatj+7hatk`
`OB=hati-6hatj+10hatk`
`OC=-hati-3hatj+4hatk`
`OD=5hati-hatj+5hatk`
Now, `AB=sqrt((7-1)^(2)+(-4+6)^(2)+(7-10)^(2))`
`=sqrt(36+4+9)`
`=sqrt(49)=7`
`BC=sqrt((1+1)^(2)+(-6+3)^(2)+(10-4)^(2))`
`=sqrt(4+9+36)`
`=sqrt(49)=7`
`CD=sqrt((-1-5)^(2)+(-3+1)^(2)+(4-5)^(2))`
`=sqrt(41)`
and `DA=sqrt((5-7)^(2)+(-1+4)^(2)+(5-7)^(2))`
`=sqrt(4+9+4)`
`=sqrt(17)`
Hence, option (d) is correct.
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