Question

The position vectors of two points A and B are \( \bar{i} + 2\bar{j} + 3\bar{k} \) and \( 7\bar{i} - \bar{k} \) respectively. The point P with position vector \( -2\bar{i} + 3\bar{j} + 5\bar{k} \) is on the line AB. If the point Q is the harmonic conjugate of P, then the sum of the scalar components of the position vector of Q is

• A. 6
• B. 4
• C. 2
• D. 0

Hint:

Harmonic conjugates simply imply ratios of \( m:n \) and \( -m:n \). Finding the ratio using just one coordinate (like x) is sufficient.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:

We first find the ratio in which P divides AB. If P divides AB in ratio \( \lambda : 1 \), then the harmonic conjugate Q divides AB in ratio \( -\lambda : 1 \).
Step 2: Key Formula or Approach:

Section Formula: \( \vec{r} = \frac{m\vec{b} + n\vec{a}}{m+n} \).
Step 3: Detailed Explanation:

Let P divide AB in ratio \( k:1 \). \[ \vec{P} = \frac{k\vec{B} + \vec{A}}{k+1} \] Equating the i-components: \[ -2 = \frac{k(7) + 1}{k+1} \] \[ -2k - 2 = 7k + 1 \] \[ -3 = 9k \implies k = -\frac{1}{3} \] P divides AB externally in ratio 1:3. Therefore, Q divides AB internally in ratio 1:3 (\( k' = \frac{1}{3} \)). \[ \vec{Q} = \frac{\frac{1}{3}\vec{B} + \vec{A}}{\frac{1}{3} + 1} = \frac{\vec{B} + 3\vec{A}}{4} \] \[ \vec{Q} = \frac{(7\bar{i} - \bar{k}) + 3(\bar{i} + 2\bar{j} + 3\bar{k})}{4} \] \[ \vec{Q} = \frac{10\bar{i} + 6\bar{j} + 8\bar{k}}{4} = \frac{5}{2}\bar{i} + \frac{3}{2}\bar{j} + 2\bar{k} \] Sum of scalar components: \[ \frac{5}{2} + \frac{3}{2} + 2 = \frac{8}{2} + 2 = 4 + 2 = 6 \]
Step 4: Final Answer:

The sum is 6.