Two vertices of a triangle are at `-hati+3hatj and 2hati+5hatj` and its orthocentre is at `hati+2hatj`. Find the position vector of third vertex.

Answer

To find the position vector of the third vertex of the triangle given two vertices and the orthocenter, we can follow these steps:

### Step 1: Identify the Given Information
We are given:
- Vertex A: \(-\hat{i} + 3\hat{j}\) (which corresponds to the point \((-1, 3)\))
- Vertex B: \(2\hat{i} + 5\hat{j}\) (which corresponds to the point \((2, 5)\))
- Orthocenter O: \(\hat{i} + 2\hat{j}\) (which corresponds to the point \((1, 2)\))

Let the third vertex C be represented as \(C(x, y)\).

### Step 2: Find the Slope of Line AB
The slope of line AB can be calculated using the formula:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
For points A \((-1, 3)\) and B \((2, 5)\):
\[ \text{slope of AB} = \frac{5 - 3}{2 - (-1)} = \frac{2}{3} \]

### Step 3: Find the Equation of Line AD (Perpendicular to AB)
Since AD is perpendicular to AB, the slope of AD will be the negative reciprocal of the slope of AB:
\[ \text{slope of AD} = -\frac{3}{2} \]
Using point-slope form of the line equation:
\[ y - 2 = -\frac{3}{2}(x - 1) \]
Rearranging gives:
\[ 3x + 2y - 7 = 0 \quad \text{(Equation 1)} \]

### Step 4: Find the Slope of Line AC
Now, we need to find the slope of line AC. The slope of AC can be expressed as:
\[ \text{slope of AC} = \frac{y - 3}{x + 1} \]
Since line BF is perpendicular to AC, we can use the slope of line BF:
\[ \text{slope of BF} = \frac{5 - 2}{2 - (-1)} = \frac{3}{3} = 1 \]
Thus, the slope of AC will be:
\[ \text{slope of AC} = -1 \]
Using point-slope form again:
\[ y - 3 = -1(x + 1) \]
Rearranging gives:
\[ x + y - 2 = 0 \quad \text{(Equation 2)} \]

### Step 5: Solve the System of Equations
Now we have two equations:
1. \(3x + 2y - 7 = 0\)
2. \(x + y - 2 = 0\)

We can solve these equations simultaneously. From Equation 2, we can express \(y\) in terms of \(x\):
\[ y = 2 - x \]
Substituting \(y\) into Equation 1:
\[ 3x + 2(2 - x) - 7 = 0 \]
Simplifying:
\[ 3x + 4 - 2x - 7 = 0 \implies x - 3 = 0 \implies x = 3 \]
Substituting \(x = 3\) back into Equation 2 to find \(y\):
\[ 3 + y - 2 = 0 \implies y = -1 \]

### Step 6: Write the Position Vector of Vertex C
The coordinates of vertex C are \((3, -1)\). Therefore, the position vector of C is:
\[ C = 3\hat{i} - \hat{j} \]

### Final Answer
The position vector of the third vertex C is:
\[ \boxed{3\hat{i} - \hat{j}} \]