Question

If the position vectors of points \( A, B, C, D \) are: \[ \vec{A} = 7\hat{i} - 4\hat{j} + 7\hat{k},\quad \vec{B} = \hat{i} - 6\hat{j} + 10\hat{k},\quad \vec{C} = -\hat{i} - 3\hat{j} + 4\hat{k},\quad \vec{D} = 5\hat{i} - \hat{j} + 5\hat{k} \] Then the quadrilateral \( ABCD \) is:

• A. A parallelogram but not a rhombus
• B. A square
• C. A quadrilateral which is not a parallelogram
• D. A rectangle

Hint:

To verify if a figure is a parallelogram, compare vectors of opposite sides. Parallel and equal vectors confirm parallelogram properties.

Answer 1

The Correct Option is C

To check if \( ABCD \) is a parallelogram: - Compute vectors \( \vec{AB} \), \( \vec{CD} \) and check if they are equal. - Also check \( \vec{BC} \) and \( \vec{DA} \) If both pairs of opposite sides are not equal or not parallel, then it is not a parallelogram. \[ \vec{AB} = \vec{B} - \vec{A} = (-6\hat{i} - 2\hat{j} + 3\hat{k}) \] \[ \vec{CD} = \vec{D} - \vec{C} = (6\hat{i} + 2\hat{j} + 1\hat{k}) \Rightarrow \vec{AB} \neq \vec{CD}, \text{ nor opposite} \] Hence, \( ABCD \) is not a parallelogram.