Vector Geometry and Proof
Vector arithmetic can prove geometric facts without measuring a diagram. This lesson develops routes, ratios, parallelism, collinearity and intersections step by step.
Higher. Complete Vectors and Vector Arithmetic first. You will also use ratio and simultaneous equations.
Build a vector route
Bold letters represent vectors. If OA=a and OB=b, then travelling A→O→B gives AB=−a+b=b−a. Write the route in order, reverse signs when travelling backwards, then simplify.
Midpoints and ratios
M is the midpoint of AB, so AM=½AB=½(b−a). Travel O→A→M: OM=a+½(b−a)=½a+½b. A midpoint position vector is the average of the endpoint position vectors.
Worked example — divide in a ratio
P lies on AB with AP:PB=2:3. There are five parts altogether, so AP=⅖AB. Thus OP=a+⅖(b−a)=⅗a+⅖b. Check: when the fraction from A is zero the expression gives a; when it is one it gives b.
Always state the ratio in its given direction. AP:PB=2:3 means P is two-fifths of the way from A, not two-thirds.
Prove lines parallel
Two non-zero vectors are parallel when one is a scalar multiple of the other. A negative scalar means opposite directions. To prove two segments parallel, express their directed vectors and exhibit that multiple.
Worked example — midpoint theorem
In triangle OAB, M is the midpoint of OA and N is the midpoint of OB. Then OM=½a and ON=½b. Therefore MN=ON−OM=½b−½a=½(b−a)=½AB. This proves MN is parallel to AB and has half its length. It is a conclusion from vectors, not an assumption from the sketch.
Prove points collinear
Collinear points lie on one straight line. Show that two non-zero vectors are scalar multiples and share a point. Parallel direction alone could describe different parallel lines.
Worked example
Suppose OP=2a−b and OQ=6a−3b, with OP non-zero. Then OQ=3OP. Both start at O, so O, P and Q are collinear. Q lies in the same direction from O, three times as far away. Hence PQ=2OP and OP:PQ=1:2.
If instead OQ=−2OP, Q would lie on the opposite side of O. State the order of the points only when the scalar and positions justify it.
Find the intersection of two routes
For harder problems, express the same position vector by two routes. If a and b are non-parallel, their coefficients must match: neither vector can be replaced by a multiple of the other.
Worked example — intersecting medians
In triangle OAB, M is the midpoint of AB and N is the midpoint of OB. The lines OM and AN meet at P. Put OP=t OM and AP=s AN.
Along OM: OP=(t/2)a+(t/2)b. Along AN: OP=a+s(½b−a)=(1−s)a+(s/2)b.
Equate the a-coefficients: t/2=1−s. Equate the b-coefficients: t/2=s/2, so t=s. Substitute: t/2=1−t, giving 3t/2=1 and t=⅔. Therefore OP is two-thirds of OM, so OP:PM=2:1.
The non-parallel condition is justified because OAB is a non-degenerate triangle. Without it, comparing coefficients this way would not establish both equations.
Think Like a Mathematician
Treat the diagram as a network of journeys. If two routes share their start and finish, they have the same vector. That equality is often the equation needed to prove the result.
Common Mistakes
AB is b−a, not a−b. A ratio 2:3 gives the fraction 2/5 along the full segment. Parallel vectors need a common point to prove collinearity of their endpoints. Do not assume parallelism or a midpoint merely because the drawing suggests it.
Practice — vector geometry and proof
1. OA=a and OB=b. Express BA.
Travel B→O→A: BA=a−b.
2. M is the midpoint of AB. Express OM.
OM=OA+AM=a+½(b−a)=½a+½b.
3. AP:PB=3:1. Express OP.
P is ¾ along AB: OP=a+¾(b−a)=¼a+¾b.
4. UV=3a+6b and XY=a+2b, both non-zero. Prove UV and XY are parallel.
UV=3XY, so they have the same direction. No claim of collinearity follows without further positional information.
5. OP=a−2b and OQ=−2a+4b, with OP non-zero. Prove O,P,Q are collinear.
OQ=−2OP. They share O, so the three points lie on one line; P and Q are on opposite sides of O.
6. In triangle OAB, OM=⅓a and ON=⅓b. Prove MN is parallel to AB.
MN=ON−OM=⅓(b−a)=⅓AB. Hence parallel, with one-third the length.
7. P lies on AB with OP=⅖a+⅗b. Find AP:PB.
OP=a+⅗(b−a), so P is ⅗ along AB. AP:PB=3:2.
Frequently Asked Questions
Can a diagram alone prove parallelism?
No. Use the stated information to obtain an exact scalar multiple and write the conclusion.
Why must a and b be non-parallel when comparing coefficients?
If one is a multiple of the other, the same vector could be written with different coefficient pairs. Independent directions make the coefficient comparison valid.
Continue Learning
Previous: Vectors and Vector Arithmetic
Final lesson in Geometry and Measures. Revisit a topic to strengthen your understanding.