But the principal failing occurred in the sailing And the bellman, perplexed and distressed, Said he had hoped, at least, when the wind blew due East That the ship would not travel due West.
If you walk 12 m east and then 5 m north, how far and in what direction will you be from your starting point?
A bird is caught in a wind blowing east at 12 m s−1 and flies so that its speed would be 5 m s−1 north in still air. What is its actual velocity?
A sledge is being pulled by two children with forces of 12 N east and 5 N north. What single force would have the same effect?
All these situations involve vectors. A vector has size (magnitude) and direction. By contrast a scalar quantity has only magnitude. There are many vector quantities; in this book you meet four of them: displacement, velocity,
acceleration and force.
When two or more dimensions are involved, the ideas underlying vectors are very important; however, in one dimension, along a straight line, you can use scalars to solve problems involving these quantities.
Although they involve quite different situations, the three problems above can be reduced to one by using the same vector techniques for finding magnitude and direction.
The instruction ‘walk 12 m east and then 5 m north’ can be modelled mathematically using a scale diagram, as in figure 5.1. The arrowed lines AB and BC are examples of vectors.
We write the vectors as The arrow above the letters is very important as it indicates the direction of the vector. means from A to B. are examples of displacement vectors. Their lengths represent the magnitude of the displacements.
It is often more convenient to use a single letter to denote a vector. For example you might see the displacement vectors written as p and q (i.e. in bold print). When writing these vectors yourself, you should underline your letters, e.g.
You can calculate the resultant using Pythagoras’ theorem and trigonometry. In triangle ABC and
The distance from the starting point is 13 m and the direction is 067°.
A special case of a displacement is a position vector. This is the displacement of a point from the origin.
Velocity and force
The other two problems that begin this chapter are illustrated in these diagrams.
Why does the bird move in the direction DF? Think what happens in very small intervals of time.
In figure 5.4, the vector represents the equivalent (resultant) force. You know that it acts at the same point on the sledge as the children’s forces, but its magnitude and direction can be found using the triangle GHJ which is similar to the two triangles, ABC and DEF.
The same diagram does for all, you just have to supply the units. The bird travels at 13 m s−1 in the direction of 067° and one child would have the same effect as the others by pulling with a force of 13 N in the direction 067°. In most of this chapter vectors are treated in the abstract. You can then apply what you learn to different real situations.
Components of a vector
It is often convenient to write one vector in terms of two others called components.
The vector a in the diagram can be split into two components in an infinite number of ways. All you need to do is to make a one side of a triangle.
It is most sensible, however, to split vectors into components in convenient directions and these
directions are usually perpendicular. Using the given grid, a is 4 units east combined with 2 units north.
You have already used components in your work and so have met the idea of vectors. For example, the total reaction between two surfaces is often split into two components. One (friction) is opposite to the direction of possible sliding and the other (normal reaction) is perpendicular to it.
Equal vectors and parallel vectors
When two vectors, p and q, are equal then they must be equal in both magnitude and direction. If they are written in component form their components must be equal.
Thus in two dimensions, the statement p = q is the equivalent of two equations (and in three dimensions, three equations).
You will often meet parallel vectors when using Newton’s second law, as in the following example.
Adding vectors in component form
In component form, addition and subtraction of vectors is simply carried out by adding or subtracting the components of the vectors.
The magnitude and direction of vectors written in component form
At the beginning of this chapter the magnitude of a vector was found by using Pythagoras’ theorem (see page 86). The direction was given using bearings, measured clockwise from the north.
When the vectors are in an x−y plane, a mathematical convention is used for direction. Starting from the x axis, angles measured anti-clockwise are positive and angles in a clockwise direction are negative as in figure 5.12.
First draw diagrams so that you can see which lengths and acute angles to find.
The vectors in each of the diagrams have the same magnitude and using Pythagoras’ theorem, the resultants all have magnitude
The angles θ are also the same size in each diagram and can be found using
The angles the vectors make starting from the x axis specify their directions:
When α is an obtuse angle, this expression is still true. For example, when
Two forces P and Q have magnitudes 4 and 5 in the directions shown in the diagram.