A point, given by the journey to it
Fix an origin O. The position vector of a point A is the vector from O to A. It is written OA with an arrow over the two letters, or with the small letter a.
Because the journey starts at the origin, its components are the coordinates of the point. A is at (5, 2), so OA = (5, 2): 5 across and 2 up from O.
Most vectors can be drawn starting anywhere; that is what makes (5, 2) a movement rather than a place. A position vector is the one exception: it always starts at O, and so it fixes exactly one point.
The position vector OA = (5, 2) runs from the origin to the point A(5, 2).
From A to B, by way of the origin
The point B is at (2, 6), so its position vector is OB = (2, 6). To go from A to B, one route goes through the origin: back along OA from A to O, and then out along OB from O to B.
Going back along OA is the negative of OA. So the vector from A to B is AB = −OA + OB, which is written AB = OB − OA. With small letters, AB = b − a: the position vector of the end minus the position vector of the start.
OA runs out to A(5, 2), and the unlabeled arrow OB out to B(2, 6). The arrow AB runs straight from A to B.
In numbers
AB = OB − OA = (2, 6) − (5, 2) = (2 − 5, 6 − 2) = (−3, 4). From A, that is 3 to the left and 4 up.
Check it by going the long way round: OA + AB = (5, 2) + (−3, 4) = (2, 6), which is OB. Starting at A and making the movement AB does land on B.
The length of AB is the distance between the two points: .
b = (−1, 3)
Make the resultant lie along the y-axis
Read the first arrow as OA, the position vector of A(3, 1), and the gold arrow as OB, the position vector of a point B, here at (2, 4). The dashed arrow between them is AB = (2 − 3, 4 − 1) = (−1, 3). Drag B anywhere: OA + AB always lands on B, so AB is always OB − OA.
The other direction
The vector from B back to A is BA = OA − OB = (5 − 2, 2 − 6) = (3, −4). It is the negative of AB: the same length, 5, pointing the opposite way. Which point is subtracted decides the direction, so always take the end minus the start.
The point halfway
The midpoint M of AB is reached by going to A and then half of the way to B: OM . Multiplying out, , so OM , half of the sum of the two position vectors.
For A(5, 2) and B(2, 6), OM . Each component is the mean of the two coordinates.
The usual mistakes
Taking the start minus the end. OA − OB = (3, −4) is the vector from B to A, not from A to B.
Adding the position vectors. OA + OB = (7, 6) is not a journey between the points; the gap between two points is the end minus the start.
Mixing up a point and a step. (−3, 4) is the movement from A to B, not the position of B, which is (2, 6).
Two villages and two masts
In the application below, the origin is a railway station and each village is given by its position vector from it. The first mast is at the midpoint, with position vector . The second is a third of the way along the road from A, so its position vector is a plus a third of AB = b − a.
Worked example: A Phone Mast Halfway Between Two Villages, and a Second a Third of the Way Along
Question On a map with its origin O at a railway station and distances in kilometers, village A has position vector a = 29 and village B has position vector b = 143. (a) A phone mast M is built halfway between the villages. Find its position vector and its distance from the station. (b) A second mast R is built on the straight road from A to B, one third of the way from A. Find its position vector.
1.The midpoint's position vector is half the sum of the two: m = 12(a + b) = 121612 = 86.
The midpoint has half the sum of the two position vectors: m = 121612 = 86. 2.(a) M has position vector 86, and its distance from the station is √82 + 62 = √100 = 10 km.
(a) M has position vector 86 and is √82 + 62 = 10 km from the station. 3.The vector along the road from A to B is AB = b − a = 143 − 29 = 12−6.
Along the road, AB = b − a = 12−6. 4.One third of the way from A is one third of that vector: 1312−6 = 4−2.
One third of the way from A is 1312−6 = 4−2 on from A. 5.(b) r = a + 13AB = 29 + 4−2 = 67. Check: from R to B is 8−4, twice 4−2, so R is twice as far from B as from A.
(b) r = a + 13AB = 29 + 4−2 = 67.
Answer: (a) m = 86, 10 km from the station; (b) r = 67
Common mistakes
- Writing the midpoint as 12(b − a) = 6−3. That is half the step from A to B; it has to be added to a before it is a position.
- Taking 13b for the second mast. A third of the position vector of B is a third of the way from the station to B, not a third of the way from A to B.