7.1 Q-12

Question Statement

Find the position vector of the point of division of the line segments joining the following pairs of points, in the given ratios:

i. Point $C$ with position vector $2 \hat{i} - 3 \hat{j}$ and point $D$ with position vector $3 \hat{i} + 2 \hat{j}$, in the ratio 4:3.
ii. Point $E$ with position vector $5 \hat{i}$ and point $F$ with position vector $4 \hat{i} + \hat{j}$, in the ratio 2:5.

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Background and Explanation

In this problem, we are asked to find the position vector of the point dividing a line segment in a given ratio. The section formula (also known as the division formula) is used to find the position of a point dividing a line segment internally in a given ratio.

The formula for the position vector $\mathbf{r}$ of a point dividing a line segment joining points $A$ and $B$ in the ratio $m:n$ is:

$$\mathbf{r} = \frac{n \mathbf{A} + m \mathbf{B}}{m + n}$$

Where:

• $\mathbf{A}$ and $\mathbf{B}$ are the position vectors of the endpoints.
• $m:n$ is the ratio in which the point divides the line segment.

We will apply this formula to find the position vectors for the given points and ratios.

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Solution

(i) Point dividing the segment $C D$ in the ratio 4:3

1. Given:

   • Position vector of $C$: $\mathbf{a} = 2 \hat{i} - 3 \hat{j}$
   • Position vector of $D$: $\mathbf{b} = 3 \hat{i} + 2 \hat{j}$
   • Ratio: 4:3

2. Apply the section formula:
   The position vector of the point dividing $C D$ in the ratio 4:3 is given by:

$$\mathbf{r} = \frac{3 \mathbf{a} + 4 \mathbf{b}}{4 + 3}$$

3. Substitute the values:

$$\mathbf{r} = \frac{3(2 \hat{i} - 3 \hat{j}) + 4(3 \hat{i} + 2 \hat{j})}{7}$$

4. Simplify the expression:

$$\mathbf{r} = \frac{(6 \hat{i} - 9 \hat{j}) + (12 \hat{i} + 8 \hat{j})}{7}$$ $$\mathbf{r} = \frac{18 \hat{i} - 1 \hat{j}}{7}$$

5. Final result:

$$\mathbf{r} = \frac{18}{7} \hat{i} - \frac{1}{7} \hat{j}$$

Thus, the position vector of the point dividing $C D$ in the ratio 4:3 is:

$$\mathbf{r} = \frac{18}{7} \hat{i} - \frac{1}{7} \hat{j}$$

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(ii) Point dividing the segment $E F$ in the ratio 2:5

1. Given:

   • Position vector of $E$: $\mathbf{a} = 5 \hat{i}$
   • Position vector of $F$: $\mathbf{b} = 4 \hat{i} + \hat{j}$
   • Ratio: 2:5

2. Apply the section formula:
   The position vector of the point dividing $E F$ in the ratio 2:5 is given by:

$$\mathbf{r} = \frac{5 \mathbf{a} + 2 \mathbf{b}}{5 + 2}$$

3. Substitute the values:

$$\mathbf{r} = \frac{5(5 \hat{i}) + 2(4 \hat{i} + \hat{j})}{7}$$

4. Simplify the expression:

$$\mathbf{r} = \frac{(25 \hat{i}) + (8 \hat{i} + 2 \hat{j})}{7}$$ $$\mathbf{r} = \frac{33 \hat{i} + 2 \hat{j}}{7}$$

5. Final result:

$$\mathbf{r} = \frac{33}{7} \hat{i} + \frac{2}{7} \hat{j}$$

Thus, the position vector of the point dividing $E F$ in the ratio 2:5 is:

$$\mathbf{r} = \frac{33}{7} \hat{i} + \frac{2}{7} \hat{j}$$

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Key Formulas or Methods Used

• Section Formula (or Division Formula):
  The position vector $\mathbf{r}$ of a point dividing the line segment joining two points $A$ and $B$ in the ratio $m:n$ is:

$$\mathbf{r} = \frac{n \mathbf{A} + m \mathbf{B}}{m + n}$$

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Summary of Steps

1. Identify the position vectors of the two points and the ratio in which the line segment is divided.
2. Use the section formula to calculate the position vector of the dividing point.
3. Simplify the expression to get the final position vector.
4. For each part of the problem, substitute the given values into the section formula and simplify the result.