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By method, show that the quadrilateral w...

By method, show that the quadrilateral with vertices A(1,2,-1), B(8,-3,-4), C(5,-1,1),D(-2,1,4) is a parallelogram.

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Let `bar(a),bar(b),bar(c),bar(d)` be the position vectors of A,B,C,D respectively w.r.t the origin
Then `bar(a)=hat(i)+2hat(j)-hat(k)," "bar(b)=8hat(i)-3hat(j)-4hat(k)," "bar(c)=5hat(i)-4hat(j)+hat(k)`,
`bar(d)=-2hat(i)+hat(j)+4hat(k)`
Let `E(bar(e))andF(bar(f))` be midpoints of AC and respectively.
Then `bar(e)=(bar(a)+bar(c))/(2)=(1)/(2)[(bar(i)+2hat(j)-bar(k))+5hat(i)-4hat(j)+hat(k))]`
`:.bar(e)=3hat(i)-hat(j)`
`andbar(f)=(bar(b)+bar(d))/(2)=(1)/(2)[(8hat(i)-3hat(j)-4hat(k))+(-2hat(i)=hat(j)+4hat(k))]`
`:.bar(f)=3hat(i)-hat(j)`
From (a) and (2), we get, . . . .(2)
`bar(e)=fbar(f)`
This show that the midpoints of diagonals AC and BD are same `:.` the diagonals AC and BD bisect each other.
Hence, the quadrilateral ABCD is a parallelogram.
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