Dividing a line in a given ratio with vectors

Dividing a Line in a Given Ratio with Vectors

Welcome! In this chapter, we are going to look at a very specific and useful skill in Further Pure Mathematics: finding the position of a point that sits somewhere on a line between two other points. Whether it is exactly in the middle or two-thirds of the way along, vectors give us a perfect way to calculate its exact position.

This topic builds on what you have already learned about position vectors and vector addition. If you need a quick reminder, remember that the position vector of a point \(A\) is written as \(\vec{OA}\) or \(\mathbf{a}\), which tells us how to get to \(A\) starting from the origin \(O\).

1. The Concept: Splitting the Line

Imagine a straight line connecting two points, \(A\) and \(B\). We want to find the position vector of a point \(P\) that lies on this line such that it divides the line in a specific ratio, \(m : n\).

Example: If the ratio is \(1 : 2\), it means point \(P\) is closer to \(A\). The distance from \(A\) to \(P\) is 1 "part," and the distance from \(P\) to \(B\) is 2 "parts." The total number of parts in the line is \(1 + 2 = 3\).

Key Term: The ratio \(m : n\) means:
\(\vec{AP} = \frac{m}{m+n} \vec{AB}\)

2. The Vector Ratio Formula

To find the position vector \(\mathbf{p}\) (which is \(\vec{OP}\)), we use the following steps:

1. We know from vector addition that to get from \(O\) to \(P\), we can go from \(O\) to \(A\), and then from \(A\) to \(P\):
\(\vec{OP} = \vec{OA} + \vec{AP}\)

2. We know that \(\vec{AP}\) is a fraction of the whole line \(\vec{AB}\). That fraction is \(\frac{m}{m+n}\).
\(\vec{OP} = \vec{OA} + \frac{m}{m+n} \vec{AB}\)

3. Remember from our previous work on position vectors that \(\vec{AB} = \mathbf{b} - \mathbf{a}\). Substituting this in:
\(\mathbf{p} = \mathbf{a} + \frac{m}{m+n} (\mathbf{b} - \mathbf{a})\)

4. If we expand and simplify this, we get the Section Formula (which is also listed in Section 8 of your syllabus for coordinates):

The Main Formula:
\(\mathbf{p} = \frac{n\mathbf{a} + m\mathbf{b}}{m + n}\)

Don't worry if this algebra seems a bit fast! The most important thing to notice is the "cross-over": the number \(n\) (the second part of the ratio) is multiplied by \(\mathbf{a}\) (the first point), and the number \(m\) (the first part of the ratio) is multiplied by \(\mathbf{b}\) (the second point).

3. A Special Case: The Midpoint

The simplest version of this is when \(P\) is exactly in the middle of \(A\) and \(B\). In this case, the ratio is \(1 : 1\).

If we plug \(m = 1\) and \(n = 1\) into our formula:
\(\mathbf{p} = \frac{1\mathbf{a} + 1\mathbf{b}}{1 + 1} = \frac{\mathbf{a} + \mathbf{b}}{2}\)

Key Takeaway: The position vector of the midpoint is simply the average of the two end position vectors!

4. Step-by-Step Example

The Problem: Point \(A\) has position vector \(\mathbf{a} = 3\mathbf{i} + 2\mathbf{j}\) and point \(B\) has position vector \(\mathbf{b} = 8\mathbf{i} - 3\mathbf{j}\). Find the position vector of point \(P\) which divides \(AB\) in the ratio \(2 : 3\).

Step 1: Identify the values.
\(m = 2\), \(n = 3\), \(\mathbf{a} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}\), \(\mathbf{b} = \begin{pmatrix} 8 \\ -3 \end{pmatrix}\)

Step 2: Use the formula.
\(\mathbf{p} = \frac{3\mathbf{a} + 2\mathbf{b}}{2 + 3}\)

Step 3: Calculate.
\(\mathbf{p} = \frac{3\begin{pmatrix} 3 \\ 2 \end{pmatrix} + 2\begin{pmatrix} 8 \\ -3 \end{pmatrix}}{5}\)
\(\mathbf{p} = \frac{\begin{pmatrix} 9 \\ 6 \end{pmatrix} + \begin{pmatrix} 16 \\ -6 \end{pmatrix}}{5}\)
\(\mathbf{p} = \frac{\begin{pmatrix} 25 \\ 0 \end{pmatrix}}{5} = \begin{pmatrix} 5 \\ 0 \end{pmatrix}\)

The final answer: \(\mathbf{p} = 5\mathbf{i}\)

5. Common Mistakes to Avoid

• Mixing up the ratio: Always remember the "cross-over" rule. If the ratio is \(A : P : B\) as \(m : n\), then \(n\) goes with \(\mathbf{a}\) and \(m\) goes with \(\mathbf{b}\).
• Forgetting the denominator: Always divide by the total number of parts (\(m + n\)), not just one of the numbers.
• Direction matters: If a question says "the ratio \(BA\) is \(m : n\)", the starting point is \(B\). Be very careful about which point is your "first" point.

6. Quick Review Box

Did you know? This vector method works for any number of dimensions, but for your IGCSE syllabus, we focus on 2D vectors using \(\mathbf{i}\) and \(\mathbf{j}\).

Summary Checklist:
1. Ratio \(m : n\) means the line has \(m+n\) total parts.
2. Midpoint position vector \(= \frac{\mathbf{a} + \mathbf{b}}{2}\).
3. General point position vector \(= \frac{n\mathbf{a} + m\mathbf{b}}{m + n}\).
4. Vector \(\vec{AB}\) is always \(\mathbf{b} - \mathbf{a}\).