Dealing with Vectors Forces are used as an example though rules apply to any vector quantities

Representing vectors Vectors can be shown either by: A line with a length proportional to the magnitude of the vector and a direction shown by the direction of the line A number with a sign / bearing / direction / angle (and a unit!)

Adding vectors The resultant of any number of vectors (forces) acting at a point can be found using two methods: By scale drawing By calculation The resultant vector is the one single vector which can replace all the other vectors acting at a point

Example 8N 5N Resultant = -8 + 5 = -3N ie 3N to left The resultant is the sum of the two vectors (remembering about directions)

Scale diagrams The resultant of 2 or more vectors can be found by joining the vectors end to end (keeping their direction correct) The resultant magnitude and direction is given by the vector joining the starting point to the end point

Example 1 8N 5N This works for any number of vectors, acting at the same point – even ones at an angle 3N 5N 8N

Example 2 It doesn’t matter which vector is started with, the magnitude and direction of the resultant is the same

So that means…… If all these forces were acting at one point, then this system of forces would have no resultant because the forces join up end to end. These forces are in equilibrium

Perpendicular pairs of vectors resultant Magnitude and direction of the resultant can be calculated using Pythagoras Complete the rectangle and find the length and direction of the diagonal

It also works for…. Pairs of vectors not at right angles - complete the parallelogram to give the same resultant obviously Pythagoras cannot be used in this case or

Remember…. Rules apply in three dimensions Problems you are given will only be in two dimensions – ie vectors on the same plain Known as co-planar vectors

Scale diagrams Sharp pencil Straight lines using an easy scale Don’t scale the angles! Careful angle measurement Diagrams as large as possible to reduce measurement errors It doesn’t matter if the vectors cross over

Now try one! 4 forces act at a point. Their magnitude and bearings are shown below. By scale diagram, find the resultant force. F1 = 45N bearing 0 degrees F2 = 20N bearing 45 degrees F3 = 55N bearing 150 degrees F4 = 30N bearing 270 degrees

Perpendicular vectors The resultant of perpendicular vectors can be calculated using Pythagoras. Find the resultant of these 4 forces 5N 5N 8N 1N

Working backwards Because we can find the resultant of perpendicular vectors, then if we can turn non-perpendicular vectors into a series of perpendicular ones, then we can find their resultant. Can be redrawn as 2 perpendicular vectors

Using trig. Sin, cos and tan relationships can be used to work out the lengths / magnitudes of each side of the triangle F sin θ F θ F cos θ

Find the resultant of this combination of forces The resultant of perpendicular vectors can be calculated using Pythagoras. Find the resultant of these 4 forces 8N 5N 5N 37 degrees 1N