Short Answer Type
If 
are distinct non-zero vectors represented by directed lines from the origin to the points A, B, C and D respectively and if 
, then prove that ABCD is a parallelogram.
Let ABCD be the quadrilateral and O be the point of intersection of the diagonals AC and BD. Let 
. O is the midpoint of both AC and BD if and only if 
or if and only if 
i.e., if any only if 
We also know that a quadrilateral is a parallelogram if and only if its opposite sides are equal and parallel.
Hence the result follows.
Long Answer TypeShort Answer TypeIf A and B are two points in space and if they become respectively A', B' on a translation, show that 
is the sum of 
show that ABQC is a || gm and Q. therefore a fixed point.
.
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are collinear, where 
are three non-coplanar vectors. 
are collinear, where 
are three non-coplanar vectors.
