Vectors for Geometric Arguments and Proofs
Geometry and measures • Vectors
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Vector notation and basic operations
A vector denotes displacement and uses bold or arrow notation, such as AB or . Vector addition represents successive displacements and scalar multiplication represents scaling along the same direction. The zero vector represents no displacement and equality of vectors requires identical magnitude and direction. Vector operations follow algebraic rules: (a + b) + c = a + (b + c) and k(a + b) = ka + kb. These properties enable rearrangement of vector equations to isolate unknown scalar parameters for geometric interpretation.
Position vectors and points
A position vector locates a point relative to an origin, commonly written as for point A. Expressing points with position vectors converts geometric statements into vector equations about those positions. For example, point P on line AB has position vector p = a + t(b - a) for some scalar t. Using position vectors requires a fixed origin and consistent coordinate frame. Subtraction of position vectors yields the displacement vector between points and enables algebraic checks for collinearity or parallelism.
Collinearity via vectors
Three points A, B and C are collinear if and only if vectors AB and AC are parallel, so AC = k·AB for some scalar k. Expressing AC and AB in terms of position vectors produces an algebraic condition a, b, c that confirms collinearity when a single scalar satisfies both components. Cause: expressing displacements as scalar multiples. Effect: existence of a scalar parameter proves collinearity and identifies relative positions and segment ratios along the line.
Concurrency and intersection of lines
Two parametric line equations r = a + s(b - a) and r = c + t(d - c) intersect when there exist scalars s and t giving the same position vector. Solving the resulting vector equation yields the parameters that locate the intersection point. Cause: equating parametric forms. Effect: solving for scalars either provides a common point (concurrency) or shows no solution (parallel or skew lines).
Parallelogram and midpoint proofs
Vectors show that opposite sides of a parallelogram are equal: AB = DC and AD = BC. Expressing vertices by position vectors and equating displacement vectors produces algebraic verification of parallelogram properties. Cause: equality of opposite displacement vectors. Effect: direct algebraic confirmation of shape properties such as equal and parallel sides and midpoint relations via averages of position vectors.
Ratios on lines and section formula
A point dividing AB in the ratio m:n has position vector p = (n a + m b)/(m + n). Writing the point as a convex combination of the endpoints converts ratio statements into linear equations in vectors that identify exact coordinates. Cause: weighted average of endpoints. Effect: exact algebraic expression for internal and external division and immediate verification of midpoint when m = n.
Parallelism and perpendicularity
Parallel vectors satisfy a = k b for some scalar k, so checking parallelism reduces to testing proportionality of components. Perpendicularity uses the dot product: vectors a and b are perpendicular when a b = 0. Combining these tests with position vectors produces direct proofs of right angles and parallel lines. Cause: component proportionality or zero dot product. Effect: algebraic criteria determine geometric relationships without coordinate geometry diagrams.
Structure of vector proofs and limiting factors
Vector proofs follow a sequence: assign position vectors, express geometric statements as vector equations, manipulate algebraically, and translate results back into geometric conclusions. Each algebraic equality implies a geometric fact because vectors represent displacements. Limiting factors include dependency on a common origin, careful tracking of scalar parameters for existence or uniqueness claims, and the need to use the correct product (dot product for perpendicularity, scalar multiplication for ratios).
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