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.

Suggested Videos

 

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\( \vec{b} \) – \( \vec{a} \)

You can download Vector Algebra Cheat Sheet by clicking on the download button below

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.