Vectors in Geometry
Apply vectors in geometry for NSW Year 12 Mathematics Extension 1 (NESA outcome ME1-12-02). Writing each point as a position vector turns questions about triangles, quadrilaterals and circles into short calculations.
You will find the point that splits an interval in a given ratio, use a scalar multiple to check whether lines are parallel or points collinear, and use the dot product to find unknown coordinates, right angles, lengths and angles in shapes.
Theory
Vectors in geometry: write points as position vectors, then use ratio division, the parallel test \(\overrightarrow{AB}=k\,\overrightarrow{CD}\) and the dot product to find points, lengths and angles in triangles, quadrilaterals and circles. This NSW Year 12 Mathematics Extension 1 topic is NESA outcome ME1-12-02.
Position vectors turn points into algebra. Fix an origin \(O\) and write \(\overrightarrow{OA}=\mathbf{a}\); any segment is then head minus tail, \(\overrightarrow{AB}=\mathbf{b}-\mathbf{a}\). With coordinates, the point \(A(1,2)\) has position vector \(\mathbf{i}+2\mathbf{j}\).
Midpoint of \(AB\): \(\tfrac{1}{2}(\mathbf{a}+\mathbf{b})\). The point \(P\) with \(AP:PB=m:n\) has \(\overrightarrow{OP}=\dfrac{n\mathbf{a}+m\mathbf{b}}{m+n}\).
Parallel or collinear: \(\overrightarrow{AB}=k\,\overrightarrow{CD}\). Perpendicular: \(\overrightarrow{AB}\cdot\overrightarrow{CD}=0\). Length: \(|\mathbf{a}|^2=\mathbf{a}\cdot\mathbf{a}\). Angle: \(\cos\theta=\dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|\,|\mathbf{b}|}\).
NESA link. Part of the Year 12 Introduction to vectors focus area, outcome ME1-12-02, with MAO-WM-01. In Extension 1 the vectors or coordinates are given and you find, express or test something; writing general geometric proofs with vectors belongs to Extension 2.
Strategy. A point on a segment: use ratio division. Parallel or collinear: look for a scalar multiple. A right angle: set the dot product to \(0\). A length or an angle: use the magnitude or the angle formula.
How to solve vector geometry problems
- Write each point as a position vector, from its coordinates or from the given letters.
- Write each segment you need as head minus tail.
- Choose the tool: ratio division for a point on a segment, a scalar multiple for parallel or collinear, a zero dot product for a right angle, the angle formula for an angle.
- Answer the question in a sentence, for example “so \(k=3\)” or “so the points are collinear”.
| \(\overrightarrow{OP}\) | \(=\) | \(\mathbf{a}+\tfrac{3}{4}(\mathbf{b}-\mathbf{a})=\tfrac{1}{4}\mathbf{a}+\tfrac{3}{4}\mathbf{b}\) |
\(\overrightarrow{OP}=\tfrac{1}{4}\mathbf{a}+\tfrac{3}{4}\mathbf{b}\).
\(\overrightarrow{OM}=\tfrac{1}{2}(\mathbf{a}+\mathbf{b})\).
| \(\overrightarrow{OG}\) | \(=\) | \(\tfrac{2}{3}\overrightarrow{OM}=\tfrac{2}{3}\cdot\tfrac{1}{2}(\mathbf{a}+\mathbf{b})=\tfrac{1}{3}(\mathbf{a}+\mathbf{b})\) |
\(\overrightarrow{OG}=\tfrac{1}{3}(\mathbf{a}+\mathbf{b})\).
Use vectors that start at \(B\): \(\overrightarrow{BA}=-3\mathbf{i}-\mathbf{j}\), \(\overrightarrow{BC}=(k-4)\mathbf{i}+3\mathbf{j}\).
| \(\overrightarrow{BA}\cdot\overrightarrow{BC}\) | \(=\) | \(-3(k-4)+(-1)(3)=-3k+9\) |
| \(-3k+9\) | \(=\) | \(0\) |
\(k=3\), so \(C\) is \((3,5)\).
\(\overrightarrow{OB}=\mathbf{a}+\mathbf{c}=8\mathbf{i}+4\mathbf{j}\), \(\overrightarrow{AC}=\mathbf{c}-\mathbf{a}=2\mathbf{i}-4\mathbf{j}\).
| \(\overrightarrow{OB}\cdot\overrightarrow{AC}\) | \(=\) | \(8(2)+4(-4)=0\) |
The dot product is \(0\), so the diagonals are perpendicular.
Common pitfalls
Frequently asked questions
How are vectors used in geometry in Extension 1?
Points are written as position vectors, then ratio division, the parallel test and the dot product are used to find points, lengths and angles.
How do you find the point dividing AB in a given ratio?
If \(AP:PB=m:n\), then \(\overrightarrow{OP}=\dfrac{n\mathbf{a}+m\mathbf{b}}{m+n}\).
How do you check whether two lines are perpendicular?
Find a direction vector for each line and their dot product; \(0\) means perpendicular.
How do you check whether three points are collinear?
Show \(\overrightarrow{AC}=k\,\overrightarrow{AB}\). The vectors are parallel and share \(A\), so the points lie on one line.
Do I need to write general vector proofs in Extension 1?
No. Extension 1 questions give vectors or coordinates and ask you to find, express or test something. General geometric proofs with vectors are part of Extension 2.