Question

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

• A. $(2, 1, 3)$
• B. $(6, 7, 8)$
• C. $(4, 1, 3)$
• D. $(7, 7, 7)$

Answer 1

Answer: (D)

$(7, 7, 7)$

Answer 2

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.
$\therefore$ $\frac{1+x}{2}=4,\frac{3+y}{2}=5,\frac{5+z}{2}=6$
$\Rightarrow$ $x=7,y=7,z=7$