The position vector of A and B are a and b respectively, then the position vector of a point P which divides AB in the ratio `1 : 2` is

Options

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

Answer

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

### Step 1: Understand the given information
We have two points A and B with position vectors represented as **a** and **b** respectively. The point P divides the line segment AB in the ratio 1:2.

### Step 2: Identify the values of M and N
In the ratio 1:2, we can assign:
- M = 1 (the part of segment AP)
- N = 2 (the part of segment PB)

### Step 3: Apply the section formula
The position vector **r** of point P that divides the line segment AB in the ratio M:N is given by the formula:
\[ \mathbf{r} = \frac{N \mathbf{a} + M \mathbf{b}}{M + N} \]
Substituting the values of M and N:
\[ \mathbf{r} = \frac{2 \mathbf{a} + 1 \mathbf{b}}{1 + 2} \]

### Step 4: Simplify the expression
Now, we can simplify the expression:
\[ \mathbf{r} = \frac{2 \mathbf{a} + \mathbf{b}}{3} \]

### Conclusion
Thus, the position vector of point P is:
\[ \mathbf{r} = \frac{2 \mathbf{a} + \mathbf{b}}{3} \]