Vector Proofs: Parallel Lines, Collinearity, and Ratios

1. The Two Foundational Principles of Vector Proofs

All vector proofs in your IGCSE course are built on two simple, interconnected ideas. Mastering these is the key.

Principle 1: Proving Lines are Parallel

Two vectors are parallel if one is a simple scalar multiple of the other. This means their directions are identical (or exactly opposite), but their lengths may differ.

If vector $\overrightarrow{PQ}$ is parallel to vector $\overrightarrow{RS}$, then you can write:
$\overrightarrow{PQ}=k\times \overrightarrow{RS}$
where $k$ is a constant number (a scalar).

• If $k$ is positive, they point in the same direction.
• If $k$ is negative, they point in opposite directions.

For example, if $\overrightarrow{AB}=2\mathbf{a}+3\mathbf{b}$ and $\overrightarrow{CD}=4\mathbf{a}+6\mathbf{b}$, we can see that $\overrightarrow{CD}=2(2\mathbf{a}+3\mathbf{b})=2\overrightarrow{AB}$. Since $\overrightarrow{CD}$ is a scalar multiple of $\overrightarrow{AB}$ (with $k=2$), the lines AB and CD are parallel.

Principle 2: Proving Points are Collinear

"Collinear" simply means the points lie on the same straight line. To prove that three points (say, A, B, and C) are collinear, you must satisfy two conditions:

1. The vector from A to B must be parallel to the vector from B to C (i.e., $\overrightarrow{AB}=k\times \overrightarrow{BC}$).
2. The two vectors must share a common point (in this case, point B).

The common point is crucial. Without it, the lines would just be parallel and separate. The shared point ensures they connect to form a single straight line.

This next clip makes the distinction between parallel and collinear explicit, which is a common point of confusion.

To summarise these critical ideas visually:

2. A Method for Collinearity Proofs

Now let's apply this to a typical IGCSE-style "show that" question. The key is to follow a clear, logical process. This turns a complex problem into a series of manageable steps and ensures you get all the method marks.

The Method:

1. Identify your Goal: State which vectors you need to prove are parallel (e.g., to prove P, N, Q are collinear, you could aim to show $\overrightarrow{PN}$ is a multiple of $\overrightarrow{NQ}$).
2. Find the Vector Paths: Use the path-finding skills from the last lesson to find expressions for your chosen vectors in terms of the base vectors (e.g., $\mathbf{a}$ and $\mathbf{b}$). This is where most of the work happens.
3. Compare and Factorise: Manipulate one of your vector expressions (usually by factoring out a number or fraction) to show it is a scalar multiple of the other.
4. State the Conclusion: Write a concluding statement that covers both conditions:
   • "Since $\overrightarrow{PN}=k\times \overrightarrow{NQ}$, the vectors are parallel."
   • "Since they also share a common point N, the points P, N, and Q are collinear."

Let's watch this method in action.

Here is another worked example, presented in a clear handwritten format. This is a good model for how to lay out your own answers in an exam.

3. Proofs Involving Ratios and Unknowns

The principles of parallelism and collinearity are also used to solve for ratios or unknown values.

Finding Ratios

If you prove that points O, P, and B are collinear by showing $\overrightarrow{OP}=\frac{2}{5}\overrightarrow{OB}$, you have also proven something about the ratio in which P divides the line OB.

• The vector $\overrightarrow{OP}$ is $\frac{2}{5}$ of the total length.
• The remaining part, $\overrightarrow{PB}$, must be $1-\frac{2}{5}=\frac{3}{5}$ of the total length.
• Therefore, the ratio $OP:PB=\frac{2}{5}:\frac{3}{5}$, which simplifies to 2:3.

Let's look at a clear written guide that walks through this process.

Finding an Unknown Scalar (e.g., 'k')

A common exam variant gives you that three points are collinear and asks you to find an unknown value k. Here, you use the condition of collinearity to form an equation.

The Method:

1. Find two different vector paths for the line segment (e.g., find expressions for $\overrightarrow{MP}$ and $\overrightarrow{MN}$). One of these will involve the unknown k.
2. Since the points M, P, N are collinear, you know $\overrightarrow{MP}$ must be a multiple of $\overrightarrow{MN}$.
3. Compare the coefficients of your base vectors ($\mathbf{a}$ and $\mathbf{b}$). This allows you to find the scalar multiple and then set up an equation to solve for k.

This technique of comparing coefficients is a powerful strategy.

Conclusion

Today, we've transformed the abstract rules of vector arithmetic into a powerful tool for constructing geometric proofs. You now have a structured approach for the "show that" questions involving vectors, which will help you write clear, logical answers that secure method marks.

Key Takeaways:

• To prove lines are parallel: Show one vector is a scalar multiple of the other ($\vec{A}=k\vec{B}$).
• To prove points are collinear: Show the vectors connecting them are parallel AND they share a common point.
• To find a ratio: The scalar multiple gives you the fraction of the line, which can be converted into a ratio (e.g., if $\overrightarrow{OP}=\frac{1}{4}\overrightarrow{OB}$, then $OP:PB=1:3$).
• To find an unknown scalar: Use the fact of collinearity to create two vector expressions, then compare coefficients to form an equation and solve.

Preview of the Next Lesson:
This lesson concludes our module on vectors. In the next lesson, we will begin a new topic: Mensuration and Circle Properties. We will start by learning to calculate the area and perimeter of complex 2D shapes, including sectors and segments of circles.