A vector is a way of representing a quantity that has size and direction. e.g. In physics, velocity and force.
Notation
Vectors are labelled either or , 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.
e.g. 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. 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.
Inverse. The inverse of a vector is obtained by changing the signs of the components of the vector.
e.g. Inverse of
This has the effect of changing the direction of the arrow on the vector.
is
Addition.
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) is sometimes shown by two arrows. Note that the two vectors go clockwise around the triangle and the resultant goes anti-clockwise.
e.g. is shown in the diagram. |
More than two vectors can be added together.
e.g.
Download an interactive spreadsheet (Microsoft Excel) illustrating the addition of vectors.
(Windows users, right click and "Save target as..." to save the files on your computer.)
Subtraction
Vector subtraction is best done by addition of the inverse or opposite vector.
By matrices
By drawing
e.g |
Navigation
Latitude and Longitude
The Earth is very nearly a perfect sphere.
On a globe of the Earth there are two sets of lines that form a special grid which can be used to locate any point on the surface of the Earth.
Lines of latitude are circles parallel to the equator.The equator is labelled as 0° and the latitude of a place is measured in degrees and minutes (1/60th of a degree) north or south of the equator. Lines of longitude are half- circles joining the North and South Poles. They cross the equator at right angles. Greenwich, near London in England, is labelled as 0° and the longitude of a place is measured in degrees and minutes east or west of Greenwich, up to 180° each way. Auckland is 37°S, 175°E
Los Angeles is 34° N, 118°W |
Bearings
Quadrant bearings These bearings have a maximum of 90°and each one begins with either N or S, followed by the number of degrees east or west of that direction. e.g. N40°W is shown in the diagram. Whole Circle bearings These bearings use north as 0° and the angle is measured in a clockwise direction. e.g. The whole circle bearing shown in the diagram is written as 320°. |