The position vectors of the points A and B are \(\hat{\imath}+2\hat{\jmath}-\hat{k}\) and \(3\hat{\imath}-2\hat{\jmath}+5\hat{k}\) respectively. Which of the following statements are correct?
(A) The position vector of the mid-point of AB is \(2\hat{\imath}+2\hat{k}\)
(B) \(\overrightarrow{AB} = 2\hat{\imath}-4\hat{\jmath}+6\hat{k}\)
(C) \(|\overrightarrow{AB}| = 2\sqrt{7}\)
(D) The position vector of the point dividing AB externally in the ratio 3 : 1 is \(4\hat{\imath}-4\hat{\jmath}+8\hat{k}\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (B) and (D) only
Let \(\vec{a}\) and \(\vec{b}\) be the position vectors of A and B. The mid-point is
\( \displaystyle \frac{\vec{a}+\vec{b}}{2} = \frac{4\hat{\imath}+0\hat{\jmath}+4\hat{k}}{2} = 2\hat{\imath}+2\hat{k} \)
The vector joining A to B is
\(\overrightarrow{AB} = \vec{b}-\vec{a} = 2\hat{\imath}-4\hat{\jmath}+6\hat{k}\)
\(|\overrightarrow{AB}| = \sqrt{4+16+36} = \sqrt{56} = 2\sqrt{14}\)
For external division in the ratio 3 : 1:
\( \displaystyle \frac{3\vec{b}-\vec{a}}{3-1} = \frac{8\hat{\imath}-8\hat{\jmath}+16\hat{k}}{2} = 4\hat{\imath}-4\hat{\jmath}+8\hat{k} \)
So the mid-point is \(2\hat{\imath}+2\hat{k}\), \(\overrightarrow{AB} = 2\hat{\imath}-4\hat{\jmath}+6\hat{k}\), \(|\overrightarrow{AB}| = 2\sqrt{14}\) and the external point is \(4\hat{\imath}-4\hat{\jmath}+8\hat{k}\).
Checking the statements
(A) The mid-point of AB is \(2\hat{\imath}+2\hat{k}\) — Correct. \(\frac{\vec{a}+\vec{b}}{2} = 2\hat{\imath}+2\hat{k}\).
(B) \(\overrightarrow{AB} = 2\hat{\imath}-4\hat{\jmath}+6\hat{k}\) — Correct. It is \(\vec{b}-\vec{a}\).
(C) \(|\overrightarrow{AB}| = 2\sqrt{7}\) — Incorrect. \(|\overrightarrow{AB}| = \sqrt{56} = 2\sqrt{14}\).
(D) The point dividing AB externally in the ratio 3 : 1 is \(4\hat{\imath}-4\hat{\jmath}+8\hat{k}\) — Correct. \(\frac{3\vec{b}-\vec{a}}{2} = 4\hat{\imath}-4\hat{\jmath}+8\hat{k}\).