Vectors and Transformations

This lesson covers the algebra and geometry of vectors, combined and inverse transformations, and their role in proofs and similarity arguments.

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Vectors

A vector describes a displacement (movement) — it has magnitude (size) and direction.

Column vector notation: a displacement of a right and b up is written as (a, b).

Magnitude: |(a, b)| = √(a² + b²)

Example: |(3, −4)| = √(9 + 16) = 5

Operations

Operation	Rule	Example
Addition	(a, b) + (c, d) = (a+c, b+d)	(2, 3) + (−1, 5) = (1, 8)
Subtraction	(a, b) − (c, d) = (a−c, b−d)	(4, 1) − (6, −2) = (−2, 3)
Scalar multiplication	k(a, b) = (ka, kb)	3(2, −1) = (6, −3)

Position Vectors

The position vector of a point P is the vector from the origin O to P, written as p or OP.

If A has position vector a and B has position vector b, then: AB = b − a (vector from A to B)