Position vectors and midpoints
If A and B have position vectors a and b, the midpoint is (a+b)/2.
Key formulas
Language
Position vectors start at the origin. Free vectors like AB can be translated anywhere.
Example
a=(2,4), b=(6,0) → midpoint (4,2). AB=(4,−4).
Section formula
Dividing in m:n gives (na+mb)/(m+n) — the vector version of the coordinate formula.
Worked example
Worked example
Problem. Position vectors a = (2, 4) and b = (6, 0). Find the midpoint M and vector AB.
Solution. M = (a+b)/2 = (4, 2). AB = b − a = (4, −4).
Origin choice
Changing origin changes position vectors but not AB.
Review quiz
a=(1,−3), b=(5,7). Midpoint is:
Practice questions
- a=(1,−3), b=(5,7). Find midpoint and AB.
- Write AB in terms of a and b.
Study notes
Bold letters?
Printed vectors are often bold; in handwriting use arrows or underlines.