Proving with position vectors
Vectors turn many facts of geometry into short algebra. The method has three steps. Choose an origin O and give the points position vectors, such as a for A and b for B. Write each line in the figure as a vector made from those letters, using “end minus start” and fractions of a line. Then read the result off the algebra: a vector that is a multiple of another is parallel to it, and the multiplier gives the ratio of their lengths.
The midpoint
M is the point halfway from A to B. To reach it from O, walk out to A and then half of the way along AB: OM = OA .
The vector AB is the end minus the start, b − a. So OM . The position vector of the midpoint is half the sum of the position vectors of the ends.
With A at (6, 2) and B at (2, 6), OM . Check: from A, half of also reaches (6 − 2, 2 + 2) = (4, 4).
The arrows OA and OB from the origin. Walking out to A and then half of AB ends at M(4, 4), the head of OM .
AB is the diagonal of a 6 by 4 box, √(6² + 4²) = 7.21; M = (1, 1) is the average of the x-coordinates and the average of the y-coordinates, found separately
Make the box 3 by 4 and read the distance
A is at (−2, −1) and B at (4, 3), so . Drag A or B: each coordinate of M is always the mean of the two, which is taken one component at a time. The length printed at the top is |AB|, by Pythagoras on the dashed box.
A point that divides a line in a ratio
P divides AB in the ratio 2 : 1, counting from A. Cut AB into 2 + 1 = 3 equal parts: 2 of them lie between A and P, and 1 between P and B. So AP is two thirds of AB, and OP , which simplifies to .
With A at (1, 1) and B at (7, 4), AB = (6, 3) and , so P is at (1 + 4, 1 + 2) = (5, 3). Check: PB = (7 − 5, 4 − 3) = (2, 1), and AP = (4, 2) is twice PB, as the ratio 2 : 1 says.
In general, for the ratio m : n from A, AP .
P(5, 3) divides AB in the ratio 2 : 1: AP = (4, 2) is two of three equal parts, and PB = (2, 1) is the third.
Proving two lines parallel
To prove that PQ is parallel to AB, show that the vector PQ is a scalar multiple of the vector AB. A multiple keeps the direction, so the two lines run the same way. With AB = (4, 2) and PQ from (1, 4) to (3, 5), PQ .
Here is a proof that holds for every triangle. In triangle OAB, let P be the midpoint of OA and Q the midpoint of OB. Then OP and OQ , so PQ = OQ − OP .
That one line proves two facts at once: PQ is parallel to AB, because it is a multiple of it, and PQ is half as long as AB, because the multiplier is .
In triangle OAB, P(3, 1) and Q(1, 3) are the midpoints of OA and OB. PQ = (−2, 2) is half of AB = (−4, 4), so it is parallel to AB and half as long.
Parallel, or on one line
A multiple shows that two vectors have the same direction. If the two also share a point, they lie on the same straight line, and the points are collinear.
Take A(1, 1), B(3, 2) and C(7, 4). AB = (3 − 1, 2 − 1) = (2, 1) and AC = (7 − 1, 4 − 1) = (6, 3) = 3AB. The two vectors are parallel, and both start at A, so A, B and C lie on one line, with C three times as far from A as B is.
Why the algebra is a proof
Measuring one drawing with a ruler shows that the result holds for that one triangle. The vector working uses nothing but the letters a and b, which can stand for any two vectors, so it holds for every triangle at once.
The usual mistakes
Giving a + b as the midpoint. The whole sum reaches twice as far; the midpoint is half of it, .
Giving half of AB as the midpoint. is a step, not a position; it has to be added to a.
Counting the ratio from the wrong end. For 2 : 1 from A, P is of the way from A, not .
Showing that two lines have the same length to prove them parallel. Two lines of equal length can point in any directions; only a multiple proves the same direction.
Getting a sign wrong on a route. From B to C through A is back along AB and then along AC: −AB + AC.
A roof truss
In the application below, the two rafters of a roof are written as vectors from the apex A: AB = p and AC = q. The base BC is reached from B by going back to the apex and then out along the other rafter, and the tie beam joins the midpoints of the rafters. The same argument as in triangle OAB, with A in place of O, shows the tie is parallel to the base and half as long.
Worked example: A Tie Beam Joining the Midpoints of Two Rafters
Question A roof truss is a triangle ABC with its apex at A. A tie beam joins M, the midpoint of the rafter AB, to N, the midpoint of the rafter AC. Let AB = p and AC = q. (a) Prove that the tie MN is parallel to the base BC and half as long. (b) The base BC is 10 m long. A second tie joins P, the midpoint of AM, to Q, the midpoint of AN. How long is the second tie?
1.Go from B to C by way of A: BC = BA + AC = −p + q = q − p.
Go from B to C by way of A: BC = BA + AC = −p + q. 2.M is halfway along AB and N is halfway along AC, so AM = 12p and AN = 12q.
M and N are halfway along the rafters: AM = 12p and AN = 12q. 3.Go from M to N by way of A: MN = MA + AN = −12p + 12q = 12(q − p).
Go from M to N by way of A: MN = −12p + 12q = 12(q − p). 4.(a) So MN = 12BC. A scalar multiple of BC is parallel to it, and the scalar 12 makes the tie half as long as the base.
(a) MN = 12BC: a scalar multiple, so parallel, and the scalar 12 makes it half as long. 5.The tie MN is 12 × 10 = 5 m long. In triangle AMN, P and Q are the midpoints of the sides AM and AN, so part (a) applies to that triangle too: PQ is parallel to MN and half as long. (b) The second tie is 12 × 5 = 2.5 m long. Check with a truss whose corners are B(0, 0), C(10, 0) and A(4, 6) in meters: M is (2, 3), N is (7, 3), P is (3, 4.5) and Q is (5.5, 4.5), so PQ = 2.50, a quarter of 100.
MN is 12 × 10 = 5 m, and part (a) applies again in triangle AMN. (b) The second tie is 12 × 5 = 2.5 m: here PQ = 2.50.
Answer: (a) MN = 12BC, so MN is parallel to BC and half as long; (b) 2.5 m
Common mistakes
- Writing BC = p − q. Going from B to C means going back along AB, which is −p, and then along AC, which is +q.
- Measuring one drawing with a ruler and calling the result proved. A measurement shows one triangle; the vector working holds for every triangle, because it uses nothing but the two midpoints.