These notes are based on the requirements of the M1 A Level mathematics module.
In terms of knowledge, there is very little to M1 vectors. You need to know about magnitude in scalar quantities and how to find this using vectors, and also about remembering to use bearings when asked (GCSE knowledge), but other than that its pretty easy going. Here goes.
Vector and Scalar Quantities
A vector is defined as a quantity that has both MAGNITUDE (an assigned value) and DIRECTION (angular displacement, or just a simple word such as ‘south’). An example is the displacement between two places, which could be 45miles; north-east.
Scalar quantities do NOT have directions, they are simply MAGNITUDE values that do not even suggest any direction or angle. An example of this is the distance between two towns, which can be called 45 miles.
Addition of vectors
This can be done on a diagram, or using numerical values for the vectors. If you have two vectors and you wish to calculate the RESULTANT VECTOR, then you simply draw out the two existing vectors and resolve them in the horizontal and vertical directions using simple trigonometry.
If dealing with i and j vectors, then you should simply add them together in terms of i and j and then find the resultant. This is more easy because the i and j components are already at right angles to each other.
Position and relative position in vectors
A position vector (usually given in terms of i and j) indicates where something that can move it starting, with respect to a given point, often denoted O. This is the origin and is the point that all position vectors are placed respective to.
Relative position vectors are the position of one vector based on where another vector is. Its like saying how to get from one of your friends houses to another, without saying your own house, which is kind of your origin. These are given in terms of i and j and to find the position vector of one, you subtract the one you are finding its position vector relative to.
For example: A relative to B is r(A) – r(B). This will give you an answer in terms of i and j.
Vectors are useful for representing many different things. Velocity, displacement, acceleration and forces are all represented as vectors at some time during the M1 syllabus. For this reason you basically have to remember that to calculate the vector of, say acceleration, you will need to use the equation for acceleration (in the next chapter). Acceleration vector will be the change in velocity vector divided by the time. Other equations, such as F = ma can also be used in this way, with Force and acceleration providing the vectors with an unchanging mass. This is how vectors can be thrown into almost any question. Just keep a cool head and remember that i and j are the eastern and northern directions respectively.
To calculate the magnitude of any of the values below, there is a simple procedure to follow. Once you have your vector for force, acceleration or something else, you must find the numerical magnitude of the i and j component. This is done by taking the coefficients of the resultant vector i and j as a and b, respectively. To get the resultant value, R, you must square both a and b, add them, and then find the square root. This is the application of Pythagoras’ theorem. These are easy marks, do not mess up! The resultant when using i and j components must always exceed the value of i and j as individuals, otherwise you have made a mistake.
- Distance Travelled = Velocity x time
- Position of Vector = Initial Position + Distance Travelled
- Velocity = Initial Velocity + (Acceleration x time)
Originally written by RobbieC on TSR forums.