Geometric proof with vectors
Prove geometric and coordinate-geometry results using vectors in 2D and 3D; prove perpendicularity using the dot product; prove parallelism using scalar multiples; prove results about midpoints, diagonals, medians, and classic theorems such as the angle in a semicircle.
Worked examples
Proving a quadrilateral is a parallelogram
Straightforward
Problem
is a quadrilateral. , , , are the midpoints of , , , respectively. Prove is a parallelogram.
1
Write position vectors for the midpoints.
2
Compute the side vectors and .
3
Conclude from the equality of the side vectors.
Since , sides and are parallel and equal in length. Therefore is a parallelogram.
Answer
is a parallelogram, since .
Angle in a semicircle
Moderate
Problem
A circle has centre at the origin and radius . Diameter has , . is any point on the circle with . Prove .
1
Write the vectors from to and from to .
2
Compute the dot product .
3
Apply the difference of squares identity .
4
Conclude from the dot product being zero.
Since the dot product is zero, .
Answer
for any point on the circle.
Medians of a triangle are concurrent
Challenging
Problem
Triangle has position vectors , , . Show all three medians meet at .
1
Identify the median from .
The median from goes to , the midpoint of .
2
Find the point dividing in ratio from .
3
Use symmetry to extend the result to the other two medians.
By symmetry, the same calculation from (to the midpoint of ) and from (to the midpoint of ) also gives .
4
Conclude.
All three medians pass through , so they are concurrent.
Answer
All three medians meet at , so they are concurrent.
Practise
Q1·Straightforward
Points , and have position vectors , and respectively. Find the scalar such that .
Explanation
and
This confirms that is the midpoint of , and , , are collinear.
This confirms that is the midpoint of , and , , are collinear.
Q2·Straightforward
Triangle has vertices , and . Find the -coordinate of the centroid .
Explanation
The -coordinate of is .
Q3·Straightforward
Position vectors of and are and . Find , the distance from to . Round to two decimal places.
Explanation
Q4·Moderate
Let be a quadrilateral with vertices , , , having position vectors , , , . Let , , , be the midpoints of , , and respectively. Prove that is a parallelogram.
✎ Work this one through on paper — proofs are self-assessed.
Worked proof
Position vectors of the midpoints:
Compute :
Compute :
Since , the sides and are equal in length and parallel.
Therefore is a parallelogram.
**Note:** This result (Varignon's theorem) holds for any quadrilateral — including non-planar ones in 3D — because the proof uses no assumption about the shape or planarity of .
Compute :
Compute :
Since , the sides and are equal in length and parallel.
Therefore is a parallelogram.
**Note:** This result (Varignon's theorem) holds for any quadrilateral — including non-planar ones in 3D — because the proof uses no assumption about the shape or planarity of .
Q5·Moderate
Let triangle have position vectors , , . Let be the midpoint of and be the midpoint of . Prove that is parallel to and that .
✎ Work this one through on paper — proofs are self-assessed.
Worked proof
Position vectors:
Compute :
Since :
- is parallel to (they are scalar multiples of the same vector), and
- .
This is the **Midsegment Theorem**: the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
Compute :
Since :
- is parallel to (they are scalar multiples of the same vector), and
- .
This is the **Midsegment Theorem**: the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
Q6·Moderate
Let be a parallelogram where and . Using as the origin, write position vectors for , and . Prove that the diagonals and bisect each other.
✎ Work this one through on paper — proofs are self-assessed.
Worked proof
Taking as origin: , , (since is a parallelogram).
**Midpoint of :**
**Midpoint of :**
Since , both diagonals pass through the same midpoint.
Therefore the diagonals of a parallelogram bisect each other.
**Midpoint of :**
**Midpoint of :**
Since , both diagonals pass through the same midpoint.
Therefore the diagonals of a parallelogram bisect each other.
Q7·Moderate
A circle has centre at the origin and radius . Points and are the ends of a diameter. is any other point on the circle, so . Prove that .
✎ Work this one through on paper — proofs are self-assessed.
Worked proof
Let have position vector with .
Compute the dot product:
Since , the angle .
**Geometric statement:** The angle in a semicircle is always a right angle.
Compute the dot product:
Since , the angle .
**Geometric statement:** The angle in a semicircle is always a right angle.
Q8·Challenging
Triangle has position vectors , , . The median from goes to the midpoint of . Let divide in the ratio from . Show that , and hence that the three medians are concurrent.
✎ Work this one through on paper — proofs are self-assessed.
Worked proof
**Midpoint of :**
**Point dividing in ratio from :**
**By symmetry**, the point dividing the median from to the midpoint of in ratio from gives the same expression , and likewise for the median from .
All three medians pass through the same point , so they are concurrent. This point is the **centroid**.
**Point dividing in ratio from :**
**By symmetry**, the point dividing the median from to the midpoint of in ratio from gives the same expression , and likewise for the median from .
All three medians pass through the same point , so they are concurrent. This point is the **centroid**.
Q9·Challenging
Let and be vectors with . Prove that and are perpendicular.
✎ Work this one through on paper — proofs are self-assessed.
Worked proof
Using :
Since :
Therefore and are perpendicular.
**Geometric interpretation:** In a rhombus, and are the two diagonals. This result proves that the diagonals of a rhombus are always perpendicular.
Q10·Challenging
Using vectors in 3D, prove that if is perpendicular to both and , then is perpendicular to every vector of the form for any scalars .
✎ Work this one through on paper — proofs are self-assessed.
Worked proof
Given: and .
For any scalars :
Since the dot product is zero, is perpendicular to for all scalars .
**Geometric meaning:** If is perpendicular to two non-parallel vectors and , then is perpendicular to the entire plane spanned by and .
For any scalars :
Since the dot product is zero, is perpendicular to for all scalars .
**Geometric meaning:** If is perpendicular to two non-parallel vectors and , then is perpendicular to the entire plane spanned by and .
Open Math
Geometric proof with vectors
Vectors · MEX-12-05
Name:
Date:
Q1Straightforward
Points , and have position vectors , and respectively. Find the scalar such that .
Q2Straightforward
Triangle has vertices , and . Find the -coordinate of the centroid .
Q3Straightforward
Position vectors of and are and . Find , the distance from to . Round to two decimal places.
Q4Moderate
Let be a quadrilateral with vertices , , , having position vectors , , , . Let , , , be the midpoints of , , and respectively. Prove that is a parallelogram.
Q5Moderate
Let triangle have position vectors , , . Let be the midpoint of and be the midpoint of . Prove that is parallel to and that .
Q6Moderate
Let be a parallelogram where and . Using as the origin, write position vectors for , and . Prove that the diagonals and bisect each other.
Q7Moderate
A circle has centre at the origin and radius . Points and are the ends of a diameter. is any other point on the circle, so . Prove that .
Q8Challenging
Triangle has position vectors , , . The median from goes to the midpoint of . Let divide in the ratio from . Show that , and hence that the three medians are concurrent.
Q9Challenging
Let and be vectors with . Prove that and are perpendicular.
Q10Challenging
Using vectors in 3D, prove that if is perpendicular to both and , then is perpendicular to every vector of the form for any scalars .
Worked solutions and answers at openmath.au/year-12/extension-2/further-work-with-vectors/geometric-proof-with-vectors