Year 11 Extended · Unit 7 Networks and vectors · Lesson 6 of 8

How do you describe where a point is, and the point halfway between two others?

A position vector is the movement from the origin to a point. With position vectors the vector between two points is the end minus the start, the midpoint is the mean of the two, and a point a given fraction of the way along is a start plus that fraction of the vector.

Geometry and TrigonometryNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From lesson 4: find the vector from (1, 2) to (5, 5).
    Answer(4, 3).
  2. From Year 7: what is the mean of 6 and 14?
    Answer10.
  3. From lesson 5: given a = (2, 1) and b = (1, 3), find a + 2b.
    Answer(4, 7).
Words used in this lesson
position vector
the vector from the origin to a point. For example: OA for the point A.
midpoint
the point halfway between two points. For example: the middle of a road.
ratio
a comparison of two parts of a whole. For example: 1 : 3.
parallelogram
a four-sided shape with opposite sides parallel. For example: a leaning rectangle.
origin
the point (0, 0). For example: where the axes meet.

Where should the hikers stop for lunch?

Two hiking groups in the Atlas Mountains set out from huts that sit at map points A and B. They agree to meet halfway. To find that point without drawing, we describe every point by a position vector: the vector from the origin O to the point, written OA or a for the point A. For the point A(−2, 1) the position vector is −2
1, and the entries of the position vector are the coordinates.

Predict first: if A is (−2, 1) and B is (6, 5), what are the coordinates of the halfway point, and how could you find them without drawing? Then use the figure to move B and a dividing point and see how they are connected.

6
5
1
(8, 4)vector AB
(0, 2)position vector of P

B = (6, 5): AB = (8, 4), and the point 1 quarter of the way from A to B has position vector (0, 2).

Show me the steps
  1. B = (6, 5): AB = (6, 5) − (−2, 1) = (8, 4). With 0 quarters P is at A, and with 4 quarters P is at B.
  2. Two quarters is the midpoint: (−2, 1) + 2 quarters of (8, 4) = (−2, 1) + (4, 2) = (2, 3), which is also the mean of the position vectors of A and B.
  3. One quarter: (−2, 1) + 1 quarter of (8, 4) = (−2, 1) + (2, 1) = (0, 2).
  4. Any B: P = A + (k ÷ 4) AB, where k is the number of quarters.

The vector between two points is the end minus the start. AB = b − a, because going from A to B is going back to the origin from A and then out to B.

Midpoint and division points. The midpoint of AB has position vector ½(a + b), the mean of the two. A point P that is a fraction f of the way from A to B has position vector a + f(b − a). If AP : PB = m : n then f = m ÷ (m + n).

Two common slips. Mixing the entries: the x-coordinate of a midpoint comes only from the two x-coordinates, and the y from the two y. And forgetting the half in the midpoint: the sum a + b is twice as far from the origin as the midpoint.

Worked examples

1. The point A has position vector 3
−2 and B is (7, 1). Find AB.

  1. AB is b − a: across 7 − 3 = 4, up 1 − (−2) = 3.
  2. AB = 4
    3    .

2. Find the midpoint of A(2, 5) and B(8, 1).

  1. Add the position vectors: (2 + 8, 5 + 1) = (10, 6).
  2. Halve: (5, 3).

3. P lies on AB with AP : PB = 2 : 1, where A is (−1, 2) and B is (8, 5). Find P.

  1. AB = (8 − (−1), 5 − 2) = (9, 3). P is 2 ÷ 3 of the way: 2 ÷ 3 × (9, 3) = (6, 2).
  2. P = (−1, 2) + (6, 2) = (5, 4).

Your go. The first step is given. M is the midpoint of PQ, where P is (−3, 4) and M is (1, 1). Find Q.

  1. Write the midpoint rule: m = ½(p + q), so q = 2m − p.
  2. Substitute the position vectors, entry by entry.
Check the answer

2m = 2
2 and p = −3
4, so q = 5
−2. Check: the mean of −3 and 5 is 1, and the mean of 4 and −2 is 1. Q = (5, −2).

Check before you go on

1. A is the point (−5, 2) and B is the point (1, −6). What is the position vector of the midpoint of AB?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. OA = 3
−4 and OB = 7
2. What is the top entry of AB?

3. A(−2, 5) and B(6, −1) are two huts on a map of a national park. What is the y-coordinate of the midpoint of AB?

Level 2 · Building

4. A is (1, 2) and B is (7, 10). The point P is on AB with AP : PB = 1 : 3. What is the x-coordinate of P, to one decimal place?

5. M is the midpoint of AB, where A is (2, −3) and M is (5, 4). What is the x-coordinate of B?

Level 3 · Stretch

6. ABCD is a parallelogram with vertices in this order and A(1, 2), B(6, 3), C(9, 8). What is the x-coordinate of D?

7. A is (3, k) and B is (9, 10). The midpoint of AB lies on the line y = x. What is k?

Something to think about

A point can be outside the segment AB on the same line. If P = A + 2 × AB, where is P, and what is the ratio AP : PB in terms of lengths?

One way to think about it

P is beyond B, at the same distance past B as B is from A, so AB = BP. The length AP is twice AB, and P is on the line through A and B but not between them. So AP : PB = 2 : 1, the same ratio as a point two thirds of the way along, but on the other side of B. Dividing a line in a ratio uses multiples between 0 and 1, and other multiples go outside it.

Exit ticket

8. O is the origin, OA = a, OB = b and M is the midpoint of AB. Which expression gives the vector from A to M?