ABCD is a parallelogram. The position vectors of A, B and C are \(3\hat{\imath}-\hat{\jmath}+2\hat{k}\), \(\hat{\imath}+2\hat{\jmath}-\hat{k}\) and \(-\hat{\imath}+\hat{\jmath}+3\hat{k}\) respectively. The position vector of D is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\hat{\imath}-2\hat{\jmath}+6\hat{k}\)
The diagonals of a parallelogram bisect each other, so AC and BD have the same mid-point:
\( \displaystyle \frac{\vec{a}+\vec{c}}{2} = \frac{\vec{b}+\vec{d}}{2} \)
\(\vec{d} = \vec{a}+\vec{c}-\vec{b}\)
\(\vec{a}+\vec{c} = 2\hat{\imath}+0\hat{\jmath}+5\hat{k}\)
\(\vec{d} = (2-1)\hat{\imath}+(0-2)\hat{\jmath}+(5+1)\hat{k}\)
\( = \hat{\imath}-2\hat{\jmath}+6\hat{k}\)
Why the other options are incorrect
Option (1) \(5\hat{\imath}-2\hat{k}\) — Incorrect. This is \(\vec{a}+\vec{b}-\vec{c}\), the fourth vertex of the parallelogram ACBD, not ABCD.
Option (3) \(-3\hat{\imath}+4\hat{\jmath}\) — Incorrect. This is \(\vec{b}+\vec{c}-\vec{a}\), the fourth vertex of the parallelogram ABDC.
Option (4) \(\hat{\imath}+\displaystyle\frac{5}{2}\hat{k}\) — Incorrect. This is the mid-point of AC, the point where the diagonals meet.