Vectors

A physical quantity that needs both a magnitude and a direction for its complete description is called a vector. Displacement, velocity, and force are common examples.

Representing a vector. A vector is represented geometrically by a directed line segment, or arrow, drawn to scale so that its length gives the magnitude of the vector and its orientation gives its direction. If a body's displacement takes it from point P to point Q, the vector is drawn as an arrow from P to Q; the starting point P is called the tail of the vector and the end point Q (where the arrowhead sits) is called its head. Symbolically this displacement vector is written $\vec{PQ}$. A vector may also be written using a single capital letter with an arrow above it, e.g. $\vec{A}$, $\vec{X}$, and so on; the magnitude of a vector $\vec{X}$ is written $|\vec{X}|$.

Types of vectors.

1. Zero vector (null vector): a vector of zero magnitude, with an arbitrary (undefined) direction, written $\vec{0}$. For example, the velocity vector of a stationary particle is a zero vector, and so is the acceleration vector of an object moving with uniform (constant) velocity.
2. Resultant vector: the single vector that produces the same overall effect as two or more vectors acting together — i.e., the vector 'sum' of those vectors.
3. Negative vector: a vector with the same magnitude as a given vector $\vec A$ but pointing in exactly the opposite direction is called the negative of $\vec A$, written $-\vec A$.
4. Equal vectors: two vectors representing the same physical quantity are equal if and only if they have the same magnitude AND the same direction. Equal vectors are drawn as parallel arrows of the same length pointing the same way.
5. Position vector: the vector that gives the position of a particle relative to the origin of a chosen coordinate system. If a particle is at point P and O is the origin, its position vector is $\vec{OP}$.
6. Unit vector: a vector of unit magnitude pointing along a given direction. If $\vec M$ is any non-zero vector (so its magnitude $M = |\vec M| \ne 0$), the unit vector along $\vec M$, written $\hat u_M$, is defined as …