Question

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?

• A. The points P,Q,R and S are NOT coplanar
• B. $\frac{\vec{b}+2\vec{d}}{3}$ is the position vector of a point that divides PR internally in the ratio 5:4
• C. $\frac{\vec{b}+2\vec{d}}{3}$ is the position vector of a point that divides PR externally in the ratio 5:4
• D. The square of the magnitude of the vector $\vec{b}\times \vec{d}$ is 95

Answer 1

Answer: (B)

$\frac{\vec{b}+2\vec{d}}{3}$ is the position vector of a point that divides PR internally in the ratio 5:4

Answer 2

Given :
The position vector of point P is given by: $\vec{P} = \hat{i} + 2\hat{j} - 5\hat{k}$
The position vector of point R is given by: $\vec{R} = \frac{17}{5}\hat{i} + \frac{16}{5}\hat{j} + 7\hat{k}$
Now, let's find the position vector of the point that divides PR internally in the ratio 5:4. This can be done using the section formula:
4 M =5+45 R +4 P ​
$\vec{M} = \frac{5\left(\frac{17}{5}\hat{i} + \frac{16}{5}\hat{j} + 7\hat{k} \right) + 4\left(\hat{i} + 2\hat{j} - 5\hat{k} \right)}{9}$
$\vec{M} = \frac{17\hat{i} + 16\hat{j} + 35\hat{k} + 4\hat{i} + 8\hat{j} - 20\hat{k}}{9}$
$\vec{M} = \frac{21\hat{i} + 24\hat{j} + 15\hat{k}}{9}$
$\vec{M} = \frac{7\hat{i} + 8\hat{j} + 5\hat{k}}{3}$
This is the same as 3 b +2 d ​, which confirms that the option (B) is indeed correct. It represents a point that divides PR internally in the ratio 5:4.
So, the correct option is (B): $\frac{\vec{b}+2\vec{d}}{3}$ is the position vector of a point that divides PR internally in the ratio 5:4