Position Vectors Coplanarity and Division Ratio of Points

To determine which of the given statements is true, we analyze each option step-by-step.

1. Evaluation of Option B and C:

First, let's compute the vector $\vec{v} = \frac{\vec{b} + 2\vec{d}}{3}$: $\vec{b} = 3\hat{i} + 6\hat{j} + 3\hat{k}$ $\vec{d} = 2\hat{i} + \hat{j} + \hat{k} \implies 2\vec{d} = 4\hat{i} + 2\hat{j} + 2\hat{k}$

Adding these together: $\vec{b} + 2\vec{d} = (3+4)\hat{i} + (6+2)\hat{j} + (3+2)\hat{k} = 7\hat{i} + 8\hat{j} + 5\hat{k}$

Thus, $\vec{v} = \frac{\vec{b} + 2\vec{d}}{3} = \frac{7}{3}\hat{i} + \frac{8}{3}\hat{j} + \frac{5}{3}\hat{k}$

Now, let's find the position vector of a point that divides the line segment $PR$ internally in the ratio $5 : 4$: $\vec{r}_{\text{internal}} = \frac{5\vec{c} + 4\vec{a}}{5 + 4}$

Substituting the position vectors $\vec{a} = \hat{i} + 2\hat{j} - 5\hat{k}$ and $\vec{c} = \frac{17}{5}\hat{i} + \frac{16}{5}\hat{j} + 7\hat{k}$: $\vec{r}_{\text{internal}} = \frac{5\left(\frac{17}{5}\hat{i} + \frac{16}{5}\hat{j} + 7\hat{k}\right) + 4(\hat{i} + 2\hat{j} - 5\hat{k})}{9}$ $\vec{r}_{\text{internal}} = \frac{(17\hat{i} + 16\hat{j} + 35\hat{k}) + (4\hat{i} + 8\hat{j} - 20\hat{k})}{9}$ $\vec{r}_{\text{internal}} = \frac{21\hat{i} + 24\hat{j} + 15\hat{k}}{9} = \frac{7}{3}\hat{i} + \frac{8}{3}\hat{j} + \frac{5}{3}\hat{k}$

Since $\vec{v} = \vec{r}_{\text{internal}}$, the point corresponding to the vector $\frac{\vec{b} + 2\vec{d}}{3}$ divides $PR$ internally in the ratio $5 : 4$.

Hence, Option B is TRUE, and Option C is FALSE.

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2. Evaluation of Option A:

To test whether the points $P, Q, R,$ and $S$ are coplanar, we compute the vectors $\vec{PQ}, \vec{PR},$ and $\vec{PS}$: $\vec{PQ} = \vec{b} - \vec{a} = (3-1)\hat{i} + (6-2)\hat{j} + (3 - (-5))\hat{k} = 2\hat{i} + 4\hat{j} + 8\hat{k}$ $\vec{PR} = \vec{c} - \vec{a} = \left(\frac{17}{5}-1\right)\hat{i} + \left(\frac{16}{5}-2\right)\hat{j} + (7 - (-5))\hat{k} = \frac{12}{5}\hat{i} + \frac{6}{5}\hat{j} + 12\hat{k}$ $\vec{PS} = \vec{d} - \vec{a} = (2-1)\hat{i} + (1-2)\hat{j} + (1 - (-5))\hat{k} = \hat{i} - \hat{j} + 6\hat{k}$

Now, compute their scalar triple product $[\vec{PQ}, \vec{PR}, \vec{PS}]$: $[\vec{PQ}, \vec{PR}, \vec{PS}] = \begin{vmatrix} 2 & 4 & 8 \\ \frac{12}{5} & \frac{6}{5} & 12 \\ 1 & -1 & 6 \end{vmatrix}$

Factoring out $\frac{6}{5}$ from the second row: $[\vec{PQ}, \vec{PR}, \vec{PS}] = \frac{6}{5} \begin{vmatrix} 2 & 4 & 8 \\ 2 & 1 & 10 \\ 1 & -1 & 6 \end{vmatrix}$ $= \frac{6}{5} \left[ 2(6 - (-10)) - 4(12 - 10) + 8(-2 - 1) \right]$ $= \frac{6}{5} \left[ 2(16) - 4(2) + 8(-3) \right]$ $= \frac{6}{5} \left[ 32 - 8 - 24 \right] = 0$

Since the scalar triple product is $0$, the points $P, Q, R,$ and $S$ are coplanar.

Hence, Option A is FALSE.

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3. Evaluation of Option D:

Let's compute the vector cross product $\vec{b} \times \vec{d}$: $\vec{b} \times \vec{d} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 6 & 3 \\ 2 & 1 & 1 \end{vmatrix}$ $= \hat{i}(6 - 3) - \hat{j}(3 - 6) + \hat{k}(3 - 12) = 3\hat{i} + 3\hat{j} - 9\hat{k}$

The square of its magnitude is: $|\vec{b} \times \vec{d}|^2 = 3^2 + 3^2 + (-9)^2 = 9 + 9 + 81 = 99 \neq 95$

Hence, Option D is FALSE.

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Conclusion:

The only correct statement is (B).