A vector is a way of representing a quantity that has size and direction. e.g. The flight of a plane or the path of a cricket ball

Vectors are also used to show where a shape moves to in a translation.

Notation

Vectors are labelled eitheror , sometimes .

A vector can be represented:

By a line, the length showing the size of the quantity and the arrow showing the direction.Vectors can start anywhere on the number plane.	Vector or
By a 2 by 1 matrix or array, enclosed in brackets.		=

Length of a Vector

The length of a vector (called its magnitude) can be found using Pythagoras' Theorem.

Properties of vectors

Multiplication by a number.
A vector can be multiplied by an ordinary number (called a constant or a scalar).
Both of the components are multiplied by the number.

e.g.

Multiplying a vector by a number produces a parallel vector.

Multiplying by negative number changes the direction of the arrow on the vector.

Adding Vectors

Vectors can be added together.

By matrices. Add the corresponding elements.

e.g.

By drawing. Form a triangle. The second vector is added on to the end of the first vector. The resultant vector (labelled c) can be given two arrows. The arrows on the resultant have opposite direction to the vectors being added.

e.g.

is shown in the diagram. Note Arrows on a and b go clockwise. Arrow on resultant c goes anti-clockwise.

For practice with vector calculations and drawings -

Solving Problems using Vectors

Many types of problems, from physics to navigation can be solved by drawing a vector diagram and then using trigonometry, or even a scale diagram.

Example

A jetboat needs to sail straight across the Waikato River in Hamilton. It is able to travel at a speed of 10 km/h in still water but the river flows at 6 km/h. a. Draw a vector triangle to show this. b. What direction must the boat head in?
a. Vector diagram b.Use a scale drawing, with 1 cm : 1 km/h Using protractor on the scale drawing x = 37° (to nearest degree)

An exciting and fun practice with bearings -