Find the position vector of R, which divides the line joining two points `P(2vec(a)+vec(b))` and `Q(vec(a)-3vec(b))` externally in the ratio 1 : 2. Also show that P is the middle point of the segment RQ.

Answer

To solve the problem, we need to find the position vector of point R, which divides the line segment joining points P and Q externally in the ratio 1:2. We also need to show that point P is the midpoint of segment RQ.

### Step 1: Identify the position vectors of points P and Q.
Given:
- \( P = 2\vec{a} + \vec{b} \)
- \( Q = \vec{a} - 3\vec{b} \)

### Step 2: Use the external division formula.
The formula for the position vector \( \vec{R} \) that divides the line joining points \( P \) and \( Q \) externally in the ratio \( m:n \) is given by:
\[ \vec{R} = \frac{m\vec{P} - n\vec{Q}}{m - n} \]
Here, \( m = 2 \) and \( n = 1 \).

### Step 3: Substitute the values into the formula.
Substituting \( \vec{P} \) and \( \vec{Q} \) into the formula:
\[ \vec{R} = \frac{2(2\vec{a} + \vec{b}) - 1(\vec{a} - 3\vec{b})}{2 - 1} \]

### Step 4: Simplify the expression.
Calculating the numerator:
\[ = \frac{(4\vec{a} + 2\vec{b}) - (\vec{a} - 3\vec{b})}{1} \]
\[ = 4\vec{a} + 2\vec{b} - \vec{a} + 3\vec{b} \]
\[ = (4\vec{a} - \vec{a}) + (2\vec{b} + 3\vec{b}) \]
\[ = 3\vec{a} + 5\vec{b} \]

Thus, the position vector of \( R \) is:
\[ \vec{R} = 3\vec{a} + 5\vec{b} \]

### Step 5: Show that P is the midpoint of segment RQ.
To find the midpoint of segment RQ, we use the midpoint formula:
\[ \text{Midpoint} = \frac{\vec{R} + \vec{Q}}{2} \]
Substituting \( \vec{R} \) and \( \vec{Q} \):
\[ = \frac{(3\vec{a} + 5\vec{b}) + (\vec{a} - 3\vec{b})}{2} \]
Calculating the numerator:
\[ = \frac{(3\vec{a} + \vec{a}) + (5\vec{b} - 3\vec{b})}{2} \]
\[ = \frac{4\vec{a} + 2\vec{b}}{2} \]
\[ = 2\vec{a} + \vec{b} \]

Since \( \vec{P} = 2\vec{a} + \vec{b} \), we conclude that:
\[ \text{Midpoint of RQ} = \vec{P} \]

### Final Conclusion:
Thus, the position vector of \( R \) is \( 3\vec{a} + 5\vec{b} \) and point \( P \) is indeed the midpoint of segment \( RQ \).

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