Vector Geometry

Any vector in the plane can be written in component form as v = (vₓ, v_y), where vₓ and v_y are the horizontal and vertical components respectively. In 3D, v = (vₓ, v_y, v_z). If A = (x₁, y₁) and B = (x₂, y₂) are two points, the vector from A to B is AB = B − A = (x₂ − x₁, y₂ − y₁). The magnitude (length) is |AB| = √((x₂−x₁)² + (y₂−y₁)²). Components allow us to add vectors, scale them, and perform all algebraic operations geometrically.

Choose a fixed origin O. The position vector of a point P is the vector OP = p, pointing from O to P. If A has position vector a and B has position vector b, then the vector joining A to B is AB = b − a. This is the cornerstone identity of vector geometry: to travel from A to B, you go 'backwards' along a and 'forwards' along b. Every geometric relationship between points can be expressed using their position vectors and this subtraction rule.

If point P divides the line segment AB in the ratio m : n (internally), then the position vector of P is p = (n·a + m·b) / (m + n). Here a and b are the position vectors of A and B. When m = n = 1, this gives the midpoint formula p = (a + b)/2. For external division in ratio m : n (where P lies outside AB), the formula becomes p = (m·b − n·a) / (m − n), provided m ≠ n. The section formula is proved by expressing AP = (m/(m+n))·AB and solving for p.

A key technique is building a route between two points by travelling along known vectors. For instance, to find vector AC, you can write AC = AB + BC (the triangle law). If you know AB = b − a and BC = c − b, then AC = c − a. This 'joining' technique lets you express any vector in terms of a small set of known basis vectors. In a parallelogram OABC with OA = a and OC = c, the diagonal OB = a + c and AC = c − a. Many geometry proofs use this method to show collinearity or find ratios.

Two vectors u and v are parallel if and only if u = k·v for some scalar k. Three points A, B, C are collinear if and only if AB = k·AC for some scalar k (i.e., the vectors are parallel and share a common point A). To test collinearity: compute AB = b − a and AC = c − a, then check whether one is a scalar multiple of the other. This is a clean, algebraic test that replaces the older gradient-comparison method from coordinate geometry.

• \mathbf{p} — Position vector of P (—): The position vector of the dividing point P relative to the origin O.
• \mathbf{a} — Position vector of A (—): Position vector of the first endpoint A of the segment.
• \mathbf{b} — Position vector of B (—): Position vector of the second endpoint B of the segment.
• m : n — Division ratio (—): The ratio in which P divides AB internally; AP : PB = m : n.

Problem. Points A and B have position vectors a = (2, 1) and b = (8, 7) relative to origin O. (a) Write down the vector AB. (b) Find the position vector of the point P that divides AB in the ratio 2 : 1 internally. (c) Verify that P lies on segment AB.

Answer. P has position vector (6, 5), dividing AB in ratio 2 : 1.

• Vector Addition and Scalar Multiplication — Vector geometry relies on adding vectors (triangle law, parallelogram law) and scaling them; these operations are the building blocks of the section formula and joining vectors.
• Magnitude and Direction of a Vector — Understanding component form requires knowing how to compute the magnitude from components using the Pythagorean theorem, and how direction is encoded in the angle the vector makes with the axes.
• Midpoint and Distance Formula — The midpoint formula is the special case m = n = 1 of the section formula, and the distance formula corresponds to the magnitude of the joining vector; these concepts connect coordinate geometry to vector geometry.
• Dot Product of Vectors — After mastering vector geometry and components, the dot product extends the toolkit to measure angles between vectors and test perpendicularity — a natural next step in the study of vectors.