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.

Explanation

Let `bar(a)=i+2hat(j)-k, bar(b)=8i-3j-4k,`
`bar(c)=5i-4j+k, d=-2i+j+4k`,
Let `bar(e)=(bar(a)+bar(c))/(2)=3i-j` ...(1)
Let `bar(f)=(bar(b)+d)/(2)=3i-j` ...(2)
From equation (1) and (2), we get `bar(e)=bar(f)`
This shows that the point `E(bar(e))` is the midpoint of diagonals AC and BD . Therefore, diagonals AC and BD bisect each other at the point E(e).
`:.` The quadrilateral ABCD is a parallelogram.
Hence Proved.