The position vector of a point is its displacement from the origin. Position vectors turn geometry problems into vector algebra.
Vectors between points
If A and B have position vectors a and b, then the vector from A to B is AB = b − a. The magnitude of a vector v = (x, y) is |v| = √(x² + y²), which gives the distance between two points.
Key idea
AB = b − a; |v| = √(x² + y²); midpoint of AB = ½(a + b). Vectors are parallel when one is a scalar multiple of the other.
Parallel and collinear
Two vectors are parallel if one equals a scalar times the other. Points A, B, C are collinear when AB and AC are parallel (they share point A).
Worked example
A = (1, 2), B = (4, 6). Then AB = b − a = (4 − 1, 6 − 2) = (3, 4), and |AB| = √(3² + 4²) = √25 = 5. The midpoint of AB is ½((1, 2) + (4, 6)) = (2.5, 4).
Position vectors also give a neat way to prove that a point divides a line in a given ratio, by writing its position vector as a weighted average of the two endpoints.
Remember
- Subtract the start point from the end point: AB = b − a.
- A unit vector is the vector divided by its magnitude.