If O is origin and C is the mid - point of A (2, -1) and B ( -4, 3) . Then value of OC is

Options

A. `hati+hatj`
B. `hati-hatj`
C. `-hati+hatj`
D. `-hati-hatj`

Explanation

To solve the problem step by step, we need to find the value of the vector OC, where O is the origin and C is the midpoint of points A and B.

### Step 1: Identify the coordinates of points A and B
- Point A has coordinates \( A(2, -1) \)
- Point B has coordinates \( B(-4, 3) \)

### Step 2: Use the midpoint formula to find the coordinates of point C
The midpoint \( C \) of two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:
\[ C\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Substituting the coordinates of A and B:
\[ C\left(\frac{2 + (-4)}{2}, \frac{-1 + 3}{2}\right) \]

### Step 3: Calculate the x-coordinate of C
\[ C_x = \frac{2 - 4}{2} = \frac{-2}{2} = -1 \]

### Step 4: Calculate the y-coordinate of C
\[ C_y = \frac{-1 + 3}{2} = \frac{2}{2} = 1 \]

### Step 5: Write the coordinates of point C
Thus, the coordinates of point C are \( C(-1, 1) \).

### Step 6: Write the position vector of point C
The position vector of point C can be written as:
\[ \vec{C} = -1 \hat{i} + 1 \hat{j} \]

### Step 7: Write the position vector of the origin O
The position vector of the origin O is:
\[ \vec{O} = 0 \hat{i} + 0 \hat{j} \]

### Step 8: Find the vector OC
The vector \( \vec{OC} \) is given by:
\[ \vec{OC} = \vec{C} - \vec{O} \]
Substituting the vectors:
\[ \vec{OC} = (-1 \hat{i} + 1 \hat{j}) - (0 \hat{i} + 0 \hat{j}) = -1 \hat{i} + 1 \hat{j} \]

### Final Answer
Thus, the value of \( \vec{OC} \) is:
\[ \vec{OC} = -\hat{i} + \hat{j} \]

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