Coordinate Geometry and Vectors
Coordinate geometry and vectors links algebra to position and movement in the plane. You need straight-line calculations and confident manipulation of vectors in column form, including magnitude, addition and parallel arguments.
Core rules
- Distance between two points = √((x₂ − x₁)² + (y₂ − y₁)²).
- Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).
- Gradient m = (y₂ − y₁)/(x₂ − x₁); subtract in the same order for both coordinates.
- Equation of a line: y = mx + c, or y − y₁ = m(x − x₁) when a point and the gradient are known.
- Parallel lines have equal gradients; perpendicular gradients satisfy m₁ × m₂ = −1.
- A column vector lists horizontal movement above vertical movement; its magnitude is √(p² + q²).
- Vector AB = b − a, where a and b are position vectors. Vectors are parallel if one is a scalar multiple of the other.
Worked example
A(1, 2) and B(5, 10) are given. Find the gradient, midpoint and length of AB, and the equation of the line through A perpendicular to AB. Reasoning: gradient = (10 − 2)/(5 − 1) = 8/4 = 2. Midpoint = ((1 + 5)/2, (2 + 10)/2) = (3, 6). Length = √(4² + 8²) = √80 ≈ 8.94. The perpendicular gradient is −½, so y − 2 = −½(x − 1), which gives y = −½x + 2.5.
Traps that cost marks
- Subtracting coordinates in opposite orders for gradient and for length.
- Losing the negative sign in the perpendicular gradient.
- Computing AB as a − b instead of b − a, reversing the direction.
- Confusing distance with midpoint.
- Giving a scalar answer where a vector is required.