Let the position vectors of the points $P, Q, R$ and $S$ be $\vec{a} = \hat{i} + 2\hat{j} - 5\hat{k}$, $\vec{b} = 3\hat{i} + 6\hat{j} + 3\hat{k}$, $\vec{c} = \frac{17}{5}\hat{i} + \frac{16}{5}\hat{j} + 7\hat{k}$ and $\vec{d} = 2\hat{i} + \hat{j} + \hat{k}$, respectively. Then which of the following statements is true?
The points $P, Q, R$ and $S$ are NOT coplanar
$\frac{\vec{b} + 2\vec{d}}{3}$ is the position vector of a point which divides $PR$ internally in the ratio $5 : 4$
$\frac{\vec{b} + 2\vec{d}}{3}$ is the position vector of a point which divides $PR$ externally in the ratio $5 : 4$
The square of the magnitude of the vector $\vec{b} \times \vec{d}$ is $95$
Step-by-Step Solution
Key Concept: Using the rotation theorem (Coni's method) and properties of equilateral triangles in the complex plane.
**Step 1: Calculate the vector expressions**
Let $\vec{v} = \frac{\vec{b} + 2\vec{d}}{3}$. We have $\vec{b} = (3, 6, 3)$ and $\vec{d} = (2, 1, 1)$. Thus $\vec{v} = \frac{1}{3} ((3, 6, 3) + (4, 2, 2)) = \frac{1}{3}(7, 8, 5) = \left(\frac{7}{3}, \frac{8}{3}, \frac{5}{3}\right)$.
**Step 2: Check division of PR**
The point $P$ is $(1, 2, -5)$ and $R$ is $\left(\frac{17}{5}, \frac{16}{5}, 7\right)$. We check if $\vec{v}$ divides $PR$ internally in the ratio $5:4$. The position vector would be $\frac{5\vec{c} + 4\vec{a}}{9}$.\n$5\vec{c} = (17, 16, 35)$. $4\vec{a} = (4, 8, -20)$.\n$5\vec{c} + 4\vec{a} = (21, 24, 15)$.\nDividing by 9: $\frac{1}{9}(21, 24, 15) = \left(\frac{21}{9}, \frac{24}{9}, \frac{15}{9}\right) = \left(\frac{7}{3}, \frac{8}{3}, \frac{5}{3}\right)$.\nThis matches $\vec{v}$, so statement (B) is true.
**Step 3: Verify coplanarity (optional)**
Since $\vec{v} = \frac{\vec{b} + 2\vec{d}}{3}$, it divides $SQ$ in ratio $1:2$. Thus the lines $PR$ and $QS$ intersect at $\vec{v}$, which implies the four points are coplanar.
Correct Answer: 2