Vectors as Proofs

Vectors as Proofs

GCSE(H),

Vectors can be used as geometric proofs, by examining the relationship between lengths and co-ordinates.

A geometric proof using vectors will often require calculating the location of a point on one or more lines using given vectors. It may be required to show this point in terms of its relation to other points or vectors.

Points may be given as a ratio on a given line e.g. a straight line ABC is split by the point B in the ratio AB:BC = 3:5. Ensure that the correct fractions of vectors are known i.e. `frac(3)(8)` and `frac(5)(8)`.

When asked to prove the intersection of two lines, it is often better to describe the intersection as two different points, then to prove that each point has the same vector.

Draw vector diagrams, labelling them with all known vectors. Move along the known vectors between given points, checking that the movement is in a positive or negative vector direction.

Examples

1. ABCDEF is a regular hexagon. X is a midpoint of AF, and Y is a midpoint of ED. Prove that XY is parallel to BC.

Answer:
`vec(XY) = vec (XF) + vec(FE) + vec(EY) `

`vec(XY) = frac(1)(2)vec(AF) + vec(EF) + frac(1)(2)vec(ED)`

`vec(XY) = (frac(1)(2)bb(b)) + (bb(b) + bb(a)) + (frac(1)(2)bb(a)) `

`vec(XY) = frac(3)(2)bb(a) + frac(3)(2)bb(b)`

`vec(XY) = frac(3)(2)(bb(a) + bb(b))`

`vec(BC) = bb(b) + bb(a)`

As `vec(XY)` is a multiple of `vec(BC)`, they are parallel.

Write all the steps as a proof.

2. AOCB is a parallelogram. Point X divides AC in the ratio 3:2. Point Y divides CB in the ratio 2:3. Prove that XY is parallel to AC.

Answer: `vec(XC) = frac(2)(5)vec(AC)`

`vec(CY) = frac(2)(3)vec(CB)`

`vec(XY) = frac(2)(5)vec(AC) + frac(2)(5)vec(CB)`

`vec(XY) = frac(2)(5)bb(b) + frac(2)(5)bb(-a)`

`vec(XY) = frac(2)(5)(bb(-a) + bb(b))`

`vec(AB) = -bb(a) + bb(b)`

As `vec(XY)` is a multiple of `vec(AB)`, they are parallel.

All the steps need to be shown as a proof