If `bar(p)=hat(i)-2hat(j)+hat(k)andbar(q)=hat(i)+4hat(j)-2hat(k)` are position vectors points P and Q. find the position vector of the points R which divides segment PQ internally in the ratio 2:1.

Answer

To find the position vector of point R that divides the segment PQ internally in the ratio 2:1, we will use the section formula. Let's denote the position vectors of points P and Q as follows:

- Position vector of P, \( \bar{p} = \hat{i} - 2\hat{j} + \hat{k} \)
- Position vector of Q, \( \bar{q} = \hat{i} + 4\hat{j} - 2\hat{k} \)

### Step 1: Identify the vectors and the ratio
We need to find the position vector of R which divides the segment PQ in the ratio 2:1. Here, we can denote:
- \( m = 2 \) (the part of Q)
- \( n = 1 \) (the part of P)

### Step 2: Apply the section formula
The section formula states that if a point R divides the line segment joining points A and B in the ratio \( m:n \), then the position vector of R is given by:

\[ \bar{r} = \frac{m \bar{b} + n \bar{a}}{m+n} \]

In our case:
- \( \bar{a} = \bar{p} \) (position vector of P)
- \( \bar{b} = \bar{q} \) (position vector of Q)

Substituting the values into the formula:

\[ \bar{r} = \frac{2 \bar{q} + 1 \bar{p}}{2 + 1} \]

### Step 3: Substitute the position vectors
Now substituting \( \bar{p} \) and \( \bar{q} \):

\[ \bar{r} = \frac{2(\hat{i} + 4\hat{j} - 2\hat{k}) + 1(\hat{i} - 2\hat{j} + \hat{k})}{3} \]

### Step 4: Calculate the components
Calculating the components inside the numerator:

1. For \( \hat{i} \):
\[ 2(\hat{i}) + 1(\hat{i}) = 2\hat{i} + \hat{i} = 3\hat{i} \]

2. For \( \hat{j} \):
\[ 2(4\hat{j}) + 1(-2\hat{j}) = 8\hat{j} - 2\hat{j} = 6\hat{j} \]

3. For \( \hat{k} \):
\[ 2(-2\hat{k}) + 1(\hat{k}) = -4\hat{k} + \hat{k} = -3\hat{k} \]

Combining these results, we have:

\[ \bar{r} = \frac{3\hat{i} + 6\hat{j} - 3\hat{k}}{3} \]

### Step 5: Simplify the expression
Now, simplifying this gives:

\[ \bar{r} = \hat{i} + 2\hat{j} - \hat{k} \]

### Final Result
Thus, the position vector of point R is:

\[ \bar{r} = \hat{i} + 2\hat{j} - \hat{k} \]

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