X and Y are two points with position vectors `3vec(a)+vec(b)` and `vec(a)-3vec(b)` respectively. Write the position vector of a point Z, which divides the line segment XY in the ratio 2 : 1 externally.

Answer

To find the position vector of point Z, which divides the line segment XY in the ratio 2:1 externally, we can follow these steps:

### Step 1: Identify the position vectors of points X and Y.
- The position vector of point X is given as:
\[ \vec{X} = 3\vec{a} + \vec{b} \]
- The position vector of point Y is given as:
\[ \vec{Y} = \vec{a} - 3\vec{b} \]

### Step 2: Use the formula for external division.
For a point Z that divides the line segment XY externally in the ratio m:n, the position vector can be calculated using the formula:
\[ \vec{Z} = \frac{n\vec{X} - m\vec{Y}}{n - m} \]
In our case, m = 2 and n = 1.

### Step 3: Substitute the values into the formula.
Substituting the values of m, n, \(\vec{X}\), and \(\vec{Y}\) into the formula:
\[ \vec{Z} = \frac{1(3\vec{a} + \vec{b}) - 2(\vec{a} - 3\vec{b})}{1 - 2} \]

### Step 4: Simplify the expression.
First, simplify the numerator:
\[ \vec{Z} = \frac{3\vec{a} + \vec{b} - 2\vec{a} + 6\vec{b}}{-1} \]
Combine like terms:
\[ \vec{Z} = \frac{(3\vec{a} - 2\vec{a}) + (1\vec{b} + 6\vec{b})}{-1} = \frac{\vec{a} + 7\vec{b}}{-1} \]
This simplifies to:
\[ \vec{Z} = -\vec{a} - 7\vec{b} \]

### Final Answer:
Thus, the position vector of point Z is:
\[ \vec{Z} = -\vec{a} - 7\vec{b} \]

---