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Maths > Vector Algebra > Section Formula
Vector Algebra

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.

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Section Formula

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

section formula

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:

section formula

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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