Consider points A,B,C annd D with position vectors `7hati-4hatj+7hatk,hati-6hatj+10hatk,-1hati-3hatj+4hatk and 5hati-hatj+5hatk`, respectively. Then, ABCD is

Options

A. square
B. rhombus
C. rectangle
D. none of these

Answer

D

Explanation

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.