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

The Correct Option is B

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:
4M=5+45R+4P​
\(\vec{M} = \frac{5\left(\frac{17}{5}\hat{i} + \frac{16}{5}\hat{j} + 7\hat{k} \right) + 4\left(\hat{i} + 2\hat{j} - 5\hat{k} \right)}{9}\)
\(\vec{M} = \frac{17\hat{i} + 16\hat{j} + 35\hat{k} + 4\hat{i} + 8\hat{j} - 20\hat{k}}{9}\)
\(\vec{M} = \frac{21\hat{i} + 24\hat{j} + 15\hat{k}}{9}\)
\(\vec{M} = \frac{7\hat{i} + 8\hat{j} + 5\hat{k}}{3}\)
This is the same as 3b+2d​, 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

Answer 2

Given :
P(1, 2, -5), Q(3, 6, 3), \(R(\frac{17}{5},\ \frac{16}{5},\ 7)\), S(2, 1, 1)
\(\frac{\vec{b}+2\vec{d}}{3}=\frac{7\hat{i}+8\hat{j}+5\hat{k}}{3}\)

\(⇒\frac{17\lambda}{5}+1=\frac{7}{3}(\lambda+1)\)
\(⇒51λ+15=35λ+35\)
\(⇒16λ=20\)
\(⇒λ=\frac{5}{4}\)
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