Many quantities in geometry and physics can be characterised by a single number with associated units, such as area (sq metres) and mass (kilograms). These quantities are called scalars. Other quantities, such as velocity and acceleration additionally require the specification of a direction and point of application. These quantities are called vectors.
Vectors on a line, say the x-axis are easy to specify: all you need is the size (eg length = 3), direction (+/-) and point of application (eg starting at x = 2).
In the x-y plane, a vector is specified by its size (eg length = 3) and point of application (eg starting at the point (2,2)). However, specifying the direction is a bit more complicated. A good way to describe the direction of a vector is by two numbers: the number of units it occupies in the x-direction and the number of units it occupies in the y direction. Its direction is now \[ \frac{\text{units of y used}}{\text{units of x used}}\] which is simply the gradient.
A vector is a directed line segment drawn from a point P (called its initial point) to a point Q (called its terminal point), with P and Q being distinct points. The vector is denoted by \( \overrightarrow{PQ} \).
In the diagram above, if P(p1, p2) and Q(q1, q2) are two points, the vector \( \overrightarrow{PQ} \) can be represented as \[ \begin{align} \overrightarrow{PQ} &= \langle \text{x units used}, \text{ y units used} \rangle \\ &= \langle q1 - p1, q2 - p2 \rangle \\ &= \langle 5 - 3, 2 - 1 \rangle \\ &= \langle 2, 1 \rangle \end{align} \ \] The two numbers, x units and y units are called the components of the the vector.
Note that vectors with the same magnitude and direction but with different initial points have the same components. This means there are an infinite number of vectors for a given magnitude and direction, differing only by their initial and terminal points.
Is there a single vector which can represent all those equal vectors? The answer is yes, and is suggested by the vector \( \overrightarrow{AB} \) in the diagram above. Vectors whose initial point is the origin are called position vectors.
Unless otherwise indicated, "the vector" with a given magnitude and direction means the one whose initial point is at the origin of the coordinate system.
Thinking of vectors as starting from the origin provides a way of dealing with vectors in a standard way, since every coordinate system has an origin. But there will be times when it is convenient to consider a different initial point for a vector (for example, when adding vectors, later in the lesson).
On the printed page a vector is indicated by a boldface letter v or when writing, \( \overrightarrow{v} \).
If a vector has the same length but the opposite direction to another vector it is the negative of the first vector.
For example, in the diagram below the vectors v and w have the same length \( \sqrt{5} \). Now v is parallel to w but points in the opposite direction. So v = -w.
The vector whose components are \( \langle \text{0 }, \text{ 0} \rangle \) is called the zero vector. Note that the direction of the zero vector is not defined.
Guided Examples
O E Qs
The magnitude of a vector is the distance between the two endpoints. If P(p1, p2) and Q(q1, q2) are two points, the magnitude of \( \overrightarrow{PQ} \) is \[ \begin{align} \parallel \overrightarrow{PQ}\parallel &= \sqrt{ (q1-p1)^2 + (q2-p2)^2} \\ \end{align}\] For position vector A(a1,a2) the magnitude is \[ \begin{align} \parallel \overrightarrow{A}\parallel &= \sqrt{ (a1)^2 + (a2)^2} \\ \end{align}\]
If P = (4,2) and Q = (16,7) find the magnitude of PQ.
Addition and Subtraction
Given two vectors u and v, add them as follows
Expressed in component form this is \[ \begin{align} \textbf{u} + \textbf{v} &= \langle u1, u2 \rangle + \langle v1, v2 \rangle \\ &= \langle u1+v1, u2+v2 \rangle \end{align}\]
Find three vectors u, v, w such that u + v + w = 0.
Subtraction is just the addition of the negative of a vector.
The component form is \[ \begin{align} \textbf{u} - \textbf{v} &= \langle u1, u2 \rangle - \langle v1, v2 \rangle \\ &= \langle u1-v1, u2-v2 \rangle \end{align}\]
Find three vectors u, v, w such that u + v = -2w.
Scalar Multiplication
If c is a scalar, i.e. a number, the scalar multiple of a vector v is cv, i.e. a vector c times as long as v in the same direction.
In component form this is \[ \begin{align} c \textbf{v} &= c \times \langle v1, v2 \rangle \\ &= \langle cv1, cv2 \rangle \end{align}\] Note that \[ \begin{align} \parallel \textbf{cv}\parallel &= |c|\parallel \textbf{v}\parallel \\ \end{align}\]
Vector arithmetic has the following properties:
You can prove these by calculating with components.
Prove that c(u+v) = cu + cv using components.