Vector Geometry
Vectors describe movement with both distance and direction. ESAT questions use vectors to define paths within geometric shapes and to construct logical proofs about properties like parallel lines or midpoints, all without needing a calculator.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- To find the vector from point A to point B, use the position vectors: `AB = OB - OA`, often written as `b - a`.
- Vectors are parallel if one is a scalar multiple of the other. For example, `v` is parallel to `w` if `v = k × w` for some number `k`.
- Vectors can be added 'head-to-tail'. The path from A to C via B is represented by the vector sum `AC = AB + BC`.
- Reversing the direction of a vector negates it. For example, the vector `BA = -AB`.
- In geometric proofs, you often need to find two different vector paths to the same point and equate them, or show one vector is a multiple of another to prove parallelism.
Formulae
AB = b - a To find the vector representing the direct path from point A to point B, using their position vectors `a` (for OA) and `b` (for OB).
AC = AB + BC The 'head-to-tail' rule for vector addition. Used to find a resultant vector by following a path through an intermediate point.
v = k × w The condition for two vectors, `v` and `w`, being parallel. If you can show this relationship holds, you have proven they are parallel.
Definitions
- Vector
- A quantity that has both magnitude (size or length) and direction.
- Scalar
- A quantity that only has magnitude, like a regular number (e.g., 5, -1/2).
- Position Vector
- A vector that starts at the origin (O) and ends at a specific point (P). It's often written in lowercase, e.g., `p`.
- Column Vector
- A representation of a vector as a column of numbers, e.g., `(x, y)`, where `x` is the horizontal component and `y` is the vertical component.
Worked example
In the trapezium OABC, the vector `OA = a` and the vector `OC = c`. The side AB is parallel to OC and is twice its length. The point D is the midpoint of BC. Find the vector `OD` in terms of `a` and `c`.
- 1
First, express the vector `AB` in terms of `c`.
Since it's parallel to `OC` and twice the length, `AB = 2 × OC = 2c`.
- 2
Next, find the vector for the side `BC`.
We can construct a path from B to C:
`BC = BA + AO + OC` - 3
Substitute the known vectors:
`BC = -AB - OA + OC = -2c - a + c = -a - c` - 4
D is the midpoint of BC, so the vector `BD` is half of `BC`.
`BD = 1/2 × BC = 1/2 × (-a - c) = -a/2 - c/2` - 5
Finally, find the vector `OD`.
We can travel from O to B and then from B to D:
`OD = OB + BD` - 6 We know `OB = OA + AB = a + 2c`
Therefore, `OD = (a + 2c) + (-a/2 - c/2)`.
- 7
Simplify by combining the `a` and `c` terms:
`OD = (1 - 1/2)a + (2 - 1/2)c = (1/2)a + (3/2)c`
Answer: OD = (1/2)a + (3/2)c
Common mistakes
- ×A very common mistake is mixing up the order for the vector between two points. Remember `AB = b - a`, not `a - b`. Think 'end minus start'.
- ×Forgetting to flip the sign when reversing a vector's direction. For instance, if a path requires you to go from B to A, you must use `-AB`.
- ×Arithmetic errors with negative signs when subtracting column vectors. For `(2, -5) - (-3, 1)`, be careful to calculate `(2 - (-3), -5 - 1)`, which correctly gives `(5, -6)`.
No-calculator tips
- ✓Always draw a quick sketch of the geometric shape. This helps you visualize the vector paths and makes it much harder to make sign errors.
- ✓When manipulating column vectors, deal with the top (x) components and the bottom (y) components as two separate, simple arithmetic problems. This reduces mental load.
- ✓When scaling by a fraction, like `(2/3) × (6, -9)`, multiply by the numerator first, then divide by the denominator: `(2*6, 2*(-9)) = (12, -18)`, then `(12/3, -18/3) = (4, -6)`.