Locating points with vectors
A position vector describes a point’s location relative to the origin:
OP⃗ = p = [x, y] for point P(x, y)
Its components match the point’s coordinates. In 3D, OP⃗ = [x, y, z].
Displacement between points
The displacement vector from A to B:
AB⃗ = OB⃗ − OA⃗ = b − a
Example: A(2, 5), B(7, 1)
AB⃗ = [7 − 2, 1 − 5] = [5, −4]
Midpoint
The midpoint M of AB has position vector:
m = (a + b)/2
Example: A(2, 5), B(8, −1) → m = [5, 2] → M(5, 2)
Dividing a segment in a ratio
A point P dividing AB in the ratio m:n (from A) has position vector given by the section formula:
p = (n·a + m·b)/(m + n)
Example: A(1, 2), B(10, 8), ratio 1:2
p = (2[1, 2] + 1[10, 8])/3 = [12, 12]/3 = [4, 4]
Check: P is one-third of the way from A to B ✓.
Proving geometric facts
Position vectors make geometry proofs algebraic.
Example: Show that the diagonals of a parallelogram bisect each other.
For parallelogram OABC with OA⃗ = a and OC⃗ = c, OB⃗ = a + c.
- Midpoint of OB: (a + c)/2
- Midpoint of AC: (a + c)/2
Same point, so the diagonals bisect each other ✓.
Centroid
The centroid (balance point) of a triangle with vertices A, B, C:
g = (a + b + c)/3
Example: A(0, 0), B(6, 0), C(3, 9) → G(3, 3).
In motion
The position of a moving object at time t can be written as a position vector r(t). For example, r(t) = [2 + 3t, 1 + 4t] describes motion in a straight line.
Worked examples
Example 1: Position vector of P(−3, 4, 1)? [−3, 4, 1].
Example 2: If a = [1, 3] and b = [5, −1], find AB⃗. [4, −4].
Example 3: Midpoint of A(−2, 6) and B(4, 0)? (1, 3).
Common mistakes
- AB⃗ = a − b (it’s b − a).
- Confusing a point with a position vector in notation.
- Using the ratio backwards in the section formula.
Try it yourself
- A(3, −2), B(−1, 6). Find AB⃗.
- Midpoint of A(1, 2, 3) and B(5, 0, −1)?
- P divides A(0, 0) to B(9, 12) in the ratio 2:1. Find P.
- Centroid of A(1, 1), B(4, 1), C(1, 7)?
Answers: 1) [−4, 8] 2) (3, 1, 1) 3) (6, 8) 4) (2, 3)
Quick recap
- A position vector OP⃗ has the point’s coordinates as components.
- Displacement vector AB⃗ = b − a.
- Midpoint = (a + b)/2; the section formula divides segments in a ratio.