Vectors (Higher)

A quantity that has both magnitude (size) and direction. Examples include displacement, velocity, and force. This contrasts with a scalar, which only has magnitude (e.g., speed, mass, temperature).

Vectors are typically represented in textbooks by bold lowercase letters (e.g., a, b). When writing by hand, an arrow above the letter is used (e.g., →a, →AB). A column vector (x, y) represents a displacement of x units horizontally and y units vertically. A position vector is a vector from the origin (O) to a point (e.g., →OA or a). A displacement vector is a vector representing the displacement from one point to another (e.g., →AB).

The length or size of a vector. For a vector a = (x, y), its magnitude, denoted |a|, is calculated using Pythagoras' theorem.

To add or subtract two vectors, add or subtract their corresponding components. Graphically, for addition, place the tail of the second vector at the head of the first; the resultant vector goes from the tail of the first to the head of the second (triangle rule). For subtraction, a - b is equivalent to a + (-b), where -b has the same magnitude as b but the opposite direction. The displacement vector →AB is found by →OB - →OA, where →OA and →OB are position vectors.

Multiplying a vector by a scalar (a number) changes its magnitude but not its direction (unless the scalar is negative, which reverses the direction). Each component of the vector is multiplied by the scalar.

Two non-zero vectors a and b are parallel if one is a scalar multiple of the other. This means they have the same or opposite direction.

Three or more points A, B, C are collinear if they lie on the same straight line. This can be proven by showing that the vector →AB is a scalar multiple of →BC (or →AC), and that they share a common point (e.g., B).