11.5PureStretch

Solving Geometric Problems

Vectors give slick proofs in geometry. Using scalar multiples and the rule AB = b − a, you can prove lines are parallel, points are collinear, and split lines in given ratios — all algebraically.

30 min Video by Zeeshan Zamurred Vectors

Edexcel AS Level Maths: 11.5 Solving Geometric Problems (Vectors)Watch the full walkthrough before the notes below.

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What you'll be able to do

Prove two vectors (lines) are parallel
Show three points are collinear
Use ratios to find points along a line
Express unknown vectors in terms of given ones

Parallel and collinear

If $A B = k C D$ the lines are $parallel$ . If two vectors share a common point AND one is a scalar multiple of the other, the three points are $collinear$ (on one straight line).

A B = k B C \Rightarrow A, B, C collinear

Parallel + shared point B ⟶ a single straight line.

Tip — Collinear = parallel vectors that share a point. Always state the shared point in your conclusion.

Working with ratios

If a point $P$ divides $A B$ in the ratio $m : n$ , you can write its position vector as a weighted combination. A point that splits $A B$ in ratio $1 : 1$ is just the midpoint $\frac{1}{2} (a + b)$ .

midpoint of A B = \frac{1}{2} (a + b)

A ratio of

1 : 1

averages the position vectors.

1Average:

\frac{1}{2} ((21) + (65))

Answer

(43)

Expressing unknown vectors

In a labelled diagram, build the vector you want by travelling along known vectors (nose-to-tail), using $A B = b - a$ as needed. Then compare scalar multiples to draw conclusions.

Tip — Always travel along edges you know. There is usually more than one valid route — pick the simplest.

Formula recap

A B = k C D \Rightarrow parallel

Parallel test.

parallel + shared point \Rightarrow collinear

Collinearity.

midpoint = \frac{1}{2} (a + b)

Midpoint of AB.

Common mistakes to avoid

Claiming points are collinear just because vectors are parallel.

They must also share a common point to be collinear.

Getting the direction of travel wrong when building a route.

Use AB = b − a and reverse a vector when you travel against it.

Key takeaways

Parallel: one vector is a scalar multiple of the other.
Collinear: parallel vectors that also share a point.
Midpoint of AB = ½(a + b); ratios give weighted combinations.
Build unknown vectors by travelling along known ones.

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