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  4. If the position vectors of three consecutive vertices, of a parallelogram are veci+ vecj+ veck, veci+3 vecj+5 veck and 7 veci+9 vecj+11 veck, then the coordinates of the fourth vertex are

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Q. If the position vectors of three consecutive vertices, of a parallelogram are $ \vec{i}+\vec{j}+\vec{k}, $ $ \vec{i}+3\vec{j}+5\vec{k} $ and $ 7\vec{i}+9\vec{j}+11\vec{k}, $ then the coordinates of the fourth vertex are

KEAMKEAM 2009Vector Algebra

Solution:

Let the vertices of a parallelogram are A(1, 1, 1) B(1, 3, 5), C(7, 9, 11) and fourth vertex be D ( $ x $ , y, z) Midpoint of AC is (4, 5, 6) and midpoint of BD is $ \left( \frac{1+x}{2},\frac{3+y}{2},\frac{5+z}{2} \right) $ .
In a parallelogram midpoint of diagonals are coincide.
$ \therefore $ $ \frac{1+x}{2}=4,\frac{3+y}{2}=5,\frac{5+z}{2}=6 $
$ \Rightarrow $ $ x=7,y=7,z=7 $