Language of vectors
Posted in Maths, Core 4A vector has both a size and direction, such as velocity which has a size - the speed and a direction - e.g. forwards or backwards. A scaler quantity has, on the other hand, only a size.
Vectors are represented as straightlines with arrow heads, the length of the arrow represents the size, also known as the magnitude of the vector. The direction of the vector is the angle between the vector and the positive x-axis with an anti-clockwise direction being positive.
The magnitude of the vector is usually denoted by r and the angle by the greek letter theta. The vector can be written in magnitude-direction form like so:
Vectors can also be written in terms of other vectors, for example:
Then can be written as where x is the amount of units in the x direction and y the amount of units in the y direction.
Unit vectors
Vectors can also be written as xi + yj, where i and j are vectors of magnitude 1, called unit vectors in the x and y directions respectivly.
To find the unit vector of a vector in a given direction you must first find the magnitude of the vector and divide by that magnitude. For example to find the unit vector in the same direction of the vector 4i + 4j firstly find the magnitude of the vector. Using phthagorus, the magnitude is given by . Then divide by the original vector by this magnitude. The unit vector is hence
You can easily convert from magnitude-direction form to unit vector form using the following rule:
The modulus of a vector is its scaler part, i.e. the size/magnitude without the direction. It is denoted by the modulus signs | |. For example, the modulus of vector a is written as |a|
Equal vectors
Two vectors are equal if they have the same direction and the same magnitude.
Position vectors
A position vector is a vector which joins the origin to another point. For example the line joining the origin and point (1, 2) is the position vector or 1i + 2j
Multiplying vectors by scalars
When a vector is multiplied by a scaler its direction changes but its direction stays the same. For example if a vector, denoted by 1i + 2j, is multiplied by a scalar, 4, the result is given by 4(1i + 2j) = 4i + 8j
Adding and subtracting vectors
Adding and subtracting vectors is done by adding or subtracting their indivual components together. For example, when the vector a = 4i + 4j is added to vector b = 2i - j the result, called the resultant, is given by a + b = 4i + 2i + 4j + - j = 6i - 3j