Mathematics · Ch 10 — Vector Algebra

Section Formula

10.5.3

Section Formula

10.5.3 Section Formula

The Core Problem

Two points $P$ and $Q$ have position vectors $\vec{OP} = \vec{p}$ and $\vec{OQ} = \vec{q}$ . A point $R$ on the line joining $P$ and $Q$ has position vector $\vec{OR} = \vec{r}$ . We find $\vec{r}$ when $R$ divides the segment $PQ$ in a given ratio — either internally ( $R$ between $P$ and $Q$ ) or externally ( $R$ on the extension beyond $P$ or beyond $Q$ ).

Important

The ratio $m : n$ is always taken with $m$ and $n$ as positive scalars. For internal division, $m$ corresponds to segment $PR$ and $n$ to segment $RQ$ .

Case I: Internal Division

When $R$ lies between $P$ and $Q$ , it divides $PQ$ internally in the ratio $m : n$ :

\frac{PR}{RQ} = \frac{m}{n}

Derivation of the Position Vector

\vec{PR} = \vec{OR} - \vec{OP} = \vec{r} - \vec{p}, \qquad \vec{RQ} = \vec{OQ} - \vec{OR} = \vec{q} - \vec{r}

Since $\vec{PR}$ and $\vec{RQ}$ are collinear and point in the same direction with magnitudes in the ratio $m : n$ :

n \cdot \vec{PR} = m \cdot \vec{RQ}

n(\vec{r} - \vec{p}) = m(\vec{q} - \vec{r})

n\vec{r} - n\vec{p} = m\vec{q} - m\vec{r}

(m + n)\vec{r} = m\vec{q} + n\vec{p}

\vec{r} = \frac{m\vec{q} + n\vec{p}}{m + n}

This is the section formula for internal division.

Note

The coefficient of $\vec{p}$ is $n$ (the ratio for segment $RQ$ ) and the coefficient of $\vec{q}$ is $m$ (the ratio for $PR$ ). Mnemonic: the ratio "opposite" to a point goes with that point.

Case II: External Division

When $R$ lies on the extension of $PQ$ (beyond $Q$ or beyond $P$ ), it divides $PQ$ externally in the ratio $m : n$ :

\frac{PR}{QR} = \frac{m}{n}

Note the denominator is $QR$ , not $RQ$ , and $R$ is outside the segment $PQ$ .

Derivation of the Position Vector

Here $\vec{PR}$ and $\vec{QR}$ point in the same direction, giving:

n(\vec{r} - \vec{p}) = m(\vec{r} - \vec{q})

n\vec{r} - n\vec{p} = m\vec{r} - m\vec{q}

(n - m)\vec{r} = n\vec{p} - m\vec{q}

\vec{r} = \frac{n\vec{p} - m\vec{q}}{n - m}

This is the section formula for external division.

Tip

To remember it: take the internal formula and change the plus in the denominator to a minus, and the plus in the numerator to a minus. (Equivalently, $\vec{r} = \frac{m\vec{q} - n\vec{p}}{m - n}$ .)

Special Case: The Midpoint

When $R$ is the midpoint of $PQ$ , the ratio $m : n$ becomes $1 : 1$ . Substituting $m = n$ into the internal formula:

$\vec{r} = \frac{1 \cdot \vec{q} + 1 \cdot \vec{p}}{1 + 1} = \frac{\vec{p} + \vec{q}}{2}$ …

Figure 10.16Internal division of segment PQ by point R, with position vectors a=OP, b=OQ and r=OR drawn from origin O and the sub-segments QR=n and RP=m, illustrating the section formula for internal division.

Fig. 10.16 — Internal division of segment PQ by point R, with position vectors a=OP, b=OQ and r=OR drawn from origin O and the sub-segments QR=n and RP=m, illustrating the section formula for internal division.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What Fig 10.16 Shows

The diagram is a clean vector sketch with three labelled points — O, P, Q, and R — and the origin O placed somewhere off to the side. From O, three position vectors radiate: $\vec{a}$ to point P, $\vec{b}$ to point Q, and $\vec{r}$ to point R. The points P and Q are joined by a thin slate-coloured line segment, and R lies on that segment between P and Q. Along the segment, two distances are marked: the length QR is labelled $n$ , and the length RP is labelled $m$ . So R divides the segment PQ internally in the ratio $m : n$ , meaning that the distance from R to P is $m$ parts and from R to Q is $n$ parts.

The figure has no axes — it is a freehand vector diagram, not a coordinate plot. Its purpose is purely geometric: to show how the position vector of a point dividing a line segment can be expressed in terms of the endpoints' position vectors.

The Physical Idea

When you have two points P and Q in space, any point R on the line joining them can be described by how far it is from P relative to the whole segment. If R lies between P and Q, we say it divides the segment internally. The ratio $m:n$ tells you that if you travel from P to Q, you cover $m$ units to reach R, then $n$ more units to reach Q — or equivalently, $\frac{PR}{RQ} = \frac{m}{n}$ .

The key insight is that the position vector $\vec{r}$ of R is a weighted average of $\vec{a}$ and $\vec{b}$ , with the weights being the opposite segment lengths. This is not obvious at first glance, which is why the textbook derives it using triangle law of vector addition.

The Derivation (in brief)

From the diagram, using triangle OQR:

\vec{r} = \vec{b} + \vec{QR}

But $\vec{QR}$ is along the direction from Q to P, and its magnitude is $n$ times the unit vector along QP. Similarly, from triangle OPR:

\vec{r} = \vec{a} + \vec{PR}

Since $\vec{PR}$ and $\vec{QR}$ are in opposite directions along the same line, and the total vector from P to Q is $\vec{b} - \vec{a}$ , we can write:

\vec{PR} = \frac{m}{m+n}(\vec{b} - \vec{a}) \quad \text{and} \quad \vec{QR} = \frac{n}{m+n}(\vec{a} - \vec{b})

Substituting either into the triangle law gives the same result.

\vec{r} = \frac{m\vec{b} + n\vec{a}}{m+n}

This is the internal division formula. Every symbol:

$\vec{r}$ — position vector of the dividing point R
$\vec{a}$ — position vector of P (first endpoint)
$\vec{b}$ — position vector of Q (second endpoint)
$m$ — the part of the segment from R to P (the "near" part to P)
$n$ — the part from R to Q (the "near" part to Q)

Watch out

A common mistake is to swap the weights. Notice that the coefficient of $\vec{a}$ is $n$ , not $m$ , and the coefficient of $\vec{b}$ is $m$ . The weight attached to an endpoint's position vector is the opposite segment length — the part of the segment that is farther from that endpoint.

Special Case: Midpoint

When $m = n$ , R is the midpoint of PQ. The formula simplifies to:

\vec{r} = \frac{\vec{a} + \vec{b}}{2}

This is the simplest and most frequently used special case — the midpoint's position vector is just the average of the two endpoints' position vectors. …

Figure 10.17External division of segment PQ by point R lying beyond Q, with position vectors a=OP, b=OQ and r=OR from origin O and the dashed segment RQ=n and RP=m, illustrating the section formula for external division.

Fig. 10.17 — External division of segment PQ by point R lying beyond Q, with position vectors a=OP, b=OQ and r=OR from origin O and the dashed segment RQ=n and RP=m, illustrating the section formula for external division.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What Fig. 10.17 Shows

The figure illustrates the external division of a line segment $PQ$ by a third point $R$ . Three position vectors emanate from the origin $O$ : $\vec{OP} = \mathbf{a}$ (to point $P$ ), $\vec{OQ} = \mathbf{b}$ (to point $Q$ ), and $\vec{OR} = \mathbf{r}$ (to point $R$ ). The three points $P$ , $Q$ , and $R$ lie on a straight line, but with a crucial ordering: $R$ lies beyond $Q$ , so the collinear arrangement is $R$ – $Q$ – $P$ (read from left to right along the line). The segment $RQ$ is drawn as a dashed indigo line, and the segment $QP$ is solid. Along $QP$ , two lengths are marked: $m$ and $n$ , where $m$ corresponds to the length $PR$ and $n$ corresponds to $QR$ .

The physical idea is simple: when a point divides a segment externally, it does not lie between the two endpoints. Instead, it lies on the extension of the segment beyond one of them. Here, $R$ is beyond $Q$ , so $Q$ lies between $R$ and $P$ . The ratio of division is given as $PR : QR = m : n$ , which is the ratio of the whole segment from $P$ to $R$ to the segment from $Q$ to $R$ .

Watch out

A common mistake is to confuse the ratio $m:n$ with the distances $PR$ and $QR$ directly. In external division, the ratio is taken as $PR : QR = m : n$ , not $PQ : QR$ or $PR : PQ$ . The point $R$ is outside the segment $PQ$ , so the distances involved are from $P$ to $R$ and from $Q$ to $R$ , not from $P$ to $Q$ .

The Key Formula

The textbook leaves the derivation as an exercise, but the result is central. For two points $P$ and $Q$ with position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, the position vector $\mathbf{r}$ of the point $R$ that divides $PQ$ externally in the ratio $m:n$ (meaning $PR : QR = m : n$ ) is:

\mathbf{r} = \frac{m\mathbf{b} - n\mathbf{a}}{m - n}

Here:

$\mathbf{a}$ = position vector of $P$ (from origin $O$ )
$\mathbf{b}$ = position vector of $Q$
$m$ and $n$ are positive scalars representing the ratio $PR : QR = m : n$
$\mathbf{r}$ = position vector of $R$

Notice the minus sign in the numerator and denominator — this is the hallmark of external division. Compare with the internal division formula $\mathbf{r} = \frac{m\mathbf{b} + n\mathbf{a}}{m + n}$ , where both terms are added. The external formula can be remembered as "the vector of the point beyond which $R$ lies ( $\mathbf{b}$ , since $R$ is beyond $Q$ ) gets the positive coefficient $m$ , and the other vector ( $\mathbf{a}$ ) gets subtracted."

Tip

A quick way to derive the external formula: treat external division as internal division with a negative ratio. If $R$ divides $PQ$ externally in the ratio $m:n$ , you can think of it as dividing $PQ$ internally in the ratio $m:(-n)$ . Substituting $n \to -n$ in the internal formula $\mathbf{r} = \frac{m\mathbf{b} + n\mathbf{a}}{m + n}$ gives $\mathbf{r} = \frac{m\mathbf{b} - n\mathbf{a}}{m - n}$ .

What the Figure Teaches Visually …

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