# Section Formula

In this article, we will look at sections of a line segment. Further, we will learn the section formula, which will help us solve problems relating to position vector formula effectively. Let’s find out more.

## Section Formula

To begin with, take a look at the figure given below:

As shown above, P and Q are two points represented by position vectors $$\vec{OP}$$ and $$\vec{OQ}$$, respectively, with respect to origin O. We can divide the line segment joining the points P and Q by a third point R in two ways:

• Internally
• Externally

If we want to find the position vector $$\vec{OR}$$ for the point R with respect to the origin O, then we should take both the cases one by one.

### Case 1 – R Divides Segment PQ Internally

Take a look at Fig. 1 again. In this figure, if the point R divides $$\vec{PQ}$$ such that,

m$$\vec{RQ}$$ = n$$\vec{PR}$$ … (1)

where ‘m’ and ‘n’ are positive scalars, then we can say that R divides $$\vec{PQ}$$ internally in the ratio m:n. Now, from the triangles ORQ and OPR, we have

$$\vec{RQ}$$ = $$\vec{OQ}$$ – $$\vec{OR}$$ = $$\vec{b}$$ – $$\vec{r}$$
And, $$\vec{PR}$$ = $$\vec{OR}$$ – $$\vec{OP}$$ = $$\vec{r}$$ – $$\vec{a}$$

Therefore, replacing the values of $$\vec{RQ}$$ and $$\vec{PR}$$ in equation (1) above, we get

m($$\vec{b}$$ – $$\vec{r}$$) = n($$\vec{r}$$ – $$\vec{a}$$)
Or, $$\vec{r}$$ = $$\frac{m\vec{b} + n\vec{a}}{m + n}$$ … (2)

Hence, the position vector forula of the point R which divides PQ internally in the ratio m:n is,

$$\vec{OR}$$ = $$\frac{m\vec{b} + n\vec{a}}{m + n}$$

### Case 2 – R Divides Segment PQ Externally

Look at the figure given below:

In Fig. 2, point R divides the segment PQ externally in the ratio m:n. Hence, we can say that point Q divides PR internally in the ratio: (m – n) : n. Therefore,

$$\frac{PQ}{QR}$$ = $$\frac{(m – n)}{n}$$

Now, by using equation (2), we have

$$\vec{b}$$ = $$\frac{(m – n)\vec{r} + n\vec{a}}{(m – n) + n}$$
$$\Longleftrightarrow$$ $$\vec{b}$$ = $$\frac{(m – n)\vec{r} + n\vec{a}}{m}$$
$$\Longleftrightarrow$$ m$$\vec{b}$$ = (m – n)$$\vec{r}$$ + n$$\vec{a}$$
Or, m$$\vec{b}$$ – n$$\vec{a}$$ = (m – n)$$\vec{r}$$
∴ $$\vec{r}$$ = $$\frac{m\vec{b} – n\vec{a}}{(m – n)}$$ … (3)

### Note: If R is the Mid-Point of PQ

If R is the mid-point of PQ, then m = n. Therefore, from equation (2) above, we have

$$\vec{r}$$ = $$\frac{m\vec{b} + m\vec{a}}{m + m}$$
Or, $$\vec{r}$$ = $$\frac{m(\vec{b} + \vec{a})}{2m}$$

Therefore, $$\vec{r}$$ = $$\frac{\vec{b} + \vec{a}}{2}$$. Hence, the position vector formula of mid-point R of PQ is,

$$\vec{OR}$$ = $$\frac{\vec{b} + \vec{a}}{2}$$ … (4)

Let’s look a solved example now:

### Example 1

Consider two points P and Q with position vectors $$\vec{OP}$$ = 3$$\vec{a}$$ − 2$$\vec{b}$$ and
$$\vec{OQ}$$ = $$\vec{a}$$ + $$\vec{b}$$. Find the position vector formula of a point R which divides the line joining P and Q in the ratio 2:1, (i) internally, and (ii) externally.

Solution: Since point R divides PQ in the ratio 2:1. we have, m = 2 and n = 1

(i) R divides PQ internally

From equation (2), we have
$$\vec{r}$$ = $$\frac{m\vec{b} + n\vec{a}}{m + n}$$
∴ $$\vec{r}$$ = $$\frac{2(\vec{a} + \vec{b}) + (3\vec{a} – 2\vec{b})}{2 + 1}$$ = $$\frac{5\vec{a}}{3}$$

(ii) R divides PQ externally

From equation (3), we have
$$\vec{r}$$ = $$\frac{m\vec{b} – n\vec{a}}{(m – n)}$$
∴ $$\vec{r}$$ = $$\frac{2(\vec{a} + \vec{b}) – (3\vec{a} – 2\vec{b})}{2 – 1}$$ = 4$$\vec{b}$$ – $$\vec{a}$$

## Solved Problems for You

Question 1: Find the position vector formula of the mid point of the vector joining the points P(2, 3, 4) and Q(4, 1, –2).

Answer : The position vector of point P(2, 3, 4) = (2 – 0)$$\hat{i}$$ + (3 – 0)$$\hat{j}$$ + (4 – 0)$$\hat{k}$$
$$\vec{OP}$$ = 2$$\hat{i}$$ + 3$$\hat{j}$$ + 4$$\hat{k}$$

Similarly, the poistion vector for Q,
$$\vec{OQ}$$ = 4$$\hat{i}$$ + 1$$\hat{j}$$ – 2$$\hat{k}$$

Let R be the mid-point of PQ. There, the position vector of R
$$\vec{OR}$$ = $$\frac{\vec{OQ} + \vec{OP}}{2}$$ … from equation (4)
∴ $$\vec{OR}$$ = $$\frac{(2\hat{i} + 3\hat{j} + 4\hat{k}) + (4\hat{i} + 1\hat{j} – 2\hat{k})}{2}$$

Hence, we get,
$$\vec{OR}$$ = $$\frac{(4 + 2)\hat{i} + (3 + 1)\hat{j} + (4 – 2)\hat{k}}{2}$$ = $$\frac{6\hat{i} + 4\hat{j} + 2\hat{k}}{2}$$
∴ $$\vec{OR}$$ = $$\frac{2(3\hat{i} + 2\hat{j} + 1\hat{k})}{2}$$
Or, $$\vec{OR}$$ = 3$$\hat{i}$$ + 2$$\hat{j}$$ + $$\hat{k}$$

Question 2: What is a vector?

Answer: It refers to an object that has both a direction and a magnitude. Moreover, in geometry, we can depict a vector as a guided line segment whose length is the magnitude of the vector and with an arrow indicating the direction. A common example of a vector that has magnitude and direction is velocity.

Question 3: What is a unit vector?

Answer: It refers to a vector that has a magnitude of 1. For instance, the vector v = (1, 3) is not a unit vector. This is so because the notation represents the norms or magnitude of vector v.

Question 4: Is force a vector?

Answer: Yes, force is a vector quantity. As explained earlier, a vector quantity is that quantity that has both a direction and a magnitude. In addition, for completelydefining the power acting upon an object, you must describe both the magnitude (numerical value or size) and the direction.

Question 5: What is the unit vector notation?

Answer: We often denote the unit vector by a lowercase letter with circumflex or “hat”(pronounced “i-hat”). Moreover, the term direction vector is used to describe a unit vector being used to represent spatial direction, and such quantities are commonly denoted as d.

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