If a and b are position vector of two points A,B and C divides AB in ratio 2:1, then position vector of C is

Options

A. `(a+2b)/(3)`
B. `(2a+b)/(3)`
C. `(a+2)/(3)`
D. `(a+b)/(2)`

Answer

To find the position vector of point C that divides the line segment AB in the ratio 2:1, we can use the section formula. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have two points A and B with position vectors represented as **a** and **b** respectively. Point C divides the line segment AB in the ratio 2:1.

### Step 2: Recall the Section Formula
The section formula states that if a point C divides the line segment joining two points A and B in the ratio m:n, then the position vector of C can be given by:
\[ \mathbf{C} = \frac{m \mathbf{b} + n \mathbf{a}}{m + n} \]

### Step 3: Assign Values to m and n
In our case, C divides AB in the ratio 2:1. Therefore, we can assign:
- \( m = 2 \) (the part towards B)
- \( n = 1 \) (the part towards A)

### Step 4: Substitute into the Section Formula
Using the section formula, we substitute the values of m and n:
\[ \mathbf{C} = \frac{2 \mathbf{b} + 1 \mathbf{a}}{2 + 1} \]

### Step 5: Simplify the Expression
Now, simplify the expression:
\[ \mathbf{C} = \frac{2 \mathbf{b} + \mathbf{a}}{3} \]
This can also be rearranged as:
\[ \mathbf{C} = \frac{\mathbf{a} + 2 \mathbf{b}}{3} \]

### Step 6: Final Result
Thus, the position vector of point C is:
\[ \mathbf{C} = \frac{\mathbf{a} + 2\mathbf{b}}{3} \]

### Conclusion
The position vector of point C that divides the line segment AB in the ratio 2:1 is:
\[ \mathbf{C} = \frac{\mathbf{a} + 2\mathbf{b}}{3} \]