Question

If a, b, c, d be the position vectors of the points A, B, C and D respectively referred to same origin O such that no three of these points are collinear and $\mathbf{x}\mathbf{i}\mathbf{-}\mathbf{j + 2k}$ then quadrilateral ABCD is a

• A. Square
• B. Rhombus
• C. Rectangle
• D. Parallelogram

Answer 1

Answer: (D)

Parallelogram

Answer 2

Given $\mathbf{a} + \mathbf{c} = \mathbf{b} + \mathbf{d} \Rightarrow \frac{1}{2}(\mathbf{a} + \mathbf{c}) = \frac{1}{2}(\mathbf{b} + \mathbf{d})$

Here, mid points of $\overset{\rightarrow}{AC}$ and $\overset{\rightarrow}{BD}$ coincide, where $\overset{\rightarrow}{AC}$ and $\overset{\rightarrow}{BD}$ are diagonals. In addition, we know that diagonals of a parallelogram bisect each other.

Hence quadrilateral is parallelogram.