84B - Vectors

In mathematics, physics and engineering, a Euclidean vector is a geometric object that has magnitude (or length) and direction and can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by a symbol with an arrow over the top.

Vectors play an important role in physics: velocity and acceleration of a moving object and forces acting on it are all described by vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example, position or displacement), their magnitude and direction can be still represented by the length and direction of an arrow. The mathematical representation of a physical vector depends on the coordinate system used to describe it.

84B1 - Vector Addition

Here you will learn to:

Represent vectors using symbols and arrows, and

Add vectors and find the magnitude and direction of the resultant vector.

Scalars

A scalar is a quantity that has magnitude (numerical size) only. Examples of scalars are the natural numbers, speed, distance, energy, charge, volume and temperature. The laws of arithmetic that apply to natural numbers can be used to manipulate scalar quantities. Many physical quantities can be added together in the same way as natural numbers. For example, if we first put 100 mL of water into a cup and then put an additional 150 mL into the cup, the cup will contain 250 mL of water. Similarly, if you were to run around a square field having a side length 100 m, you would have run a total distance of 400 m. Such quantities can also be subtracted in the usual way. For example, if you were to eat 100 g cheese from a piece of mass 500 g, the mass of the remaining piece would be 400 g.

In the examples above, we have used volume, distance and mass as examples of physical quantities called scalars. Other examples are time, temperature and any natural number. The value of a scalar is called its magnitude.

Vectors

The displacement of a walker after moving 4.0 m north
and then 3.0 m east.

A vector is a quantity that has both a magnitude and a direction. Vectors are used to describe some physical quantities that are not scalars. Examples of vectors are displacement, velocity, acceleration, force and electric field. Special arithmetic rules must be obeyed when adding vectors together which are not the same as those applied to natural numbers and much of this section is devoted to learning these rules.

Displacement is an example of a vector and is a useful example to show why vector quantities cannot be added according to the simple rules of arithmetic described for scalars. For example, if you were to walk 4 m in a northerly direction and then 3 m in an easterly direction, how far would you be from your starting point? The answer is clearly not 3 + 4 = 7 m! To find the answer, you could draw a scale diagram (1 cm = 1 m) such as the one shown on the right. We could also calculate the distance from the starting point using Pythagoras' theorem. It is also useful to know in which direction you have moved from your starting point. This can also be measured from the diagram or calculated from simple trigonometry and it tells us that you could have reached the same final position by walking 5 m in the direction 37° east of north. This is the result of adding 4 m north and 3 m east. The physical quantities, 4 m north, 3 m east and 5 m 37° east of north require both a magnitude and a direction to fully describe them. These displacement quantities are examples of vector quantities. Other examples of vector quantities that you will encounter are velocity, acceleration and force. All vector quantities can be added together in the same way as the displacements shown in the diagram.

Representing Vectors

Different ways to represent vectors.

Vectors are distinguished from scalars by writing them in special ways. A widely used convention is to draw a small arrow above the symbol for the vector (see diagram on right).

Since several important physical quantities are vectors, it is useful to agree on a way for representing them and adding them together. In the example involving displacement above, we used a scale diagram in which displacements were represented by arrows which were proportionately scaled and orientated correctly with respect to our axes (i.e., the points of the compass). This representation can be used for all vector quantities provided the following rules are followed:

  1. The reference direction is indicated (i.e. north or right),

  2. The scale is indicated,

  3. The vectors are represented as arrows with a length proportional to their magnitude and are correctly orientated with respect to the reference direction, and

  4. The direction of the vector is indicated by an arrowhead.

The arrows should be labelled to show which vectors they represent. For example, the diagram on the right shows two vectors A and B, where A has a magnitude of 3 units in a direction parallel to the reference direction and B has a magnitude of 2 units and a direction 60° clockwise to the reference direction.

Two vectors are equal when they have the same magnitude and direction, irrespective of their point of origin. In the diagram on the right, vector A is equal to vector B since they have the same magnitude and direction. A negative vector has the same magnitude but opposite direction to its corresponding positive vector and this is also demonstrated in the diagram.

Vector Addition

Vector diagrams showing how to add two (left) and three (right)
component vectors.

Vector addition involves drawing a diagram whereby arrows are drawn for each vector and they are added together one-by-one so that the head of the next vecotr is added to the tail of the previous vector. The resultant vector is then drawn as an arrow from the tail of the first vector to the head of the last vector. The order in which the vectors are added makes no difference to the final answer or resultant vector just as adding 1 + 2 + 3. The answer will always be 6 regardless of the order in which the numbers are added.

When adding two vectors together:

  1. Draw each vector to be added to scale indicating direction and label with the symbol and magnitude,

  2. Draw the first vector,

  3. Add the tail of the second vector to the head of the first vector,

  4. Draw the resultant vector as a line joining the tail of the first vector to the head of the second vector, and

  5. Add an arrow head to the resultant vector where it joins to the head of the second vector.

When two vectors not in the same dimension are added together, you will end up with a triangle. You can then use trigonometry to find the length of the resultant and the angle for the direction. If you have three or more vectors you are adding together, you will end up with a polygon of some sort. In this case the only way to calculate the magnitude and direction of the resultant vector is to draw a scale vector addition diagram, measure the length of the resultant with a ruler and the angle with a protractor.

Stuff to Do

TutorialTutorial 84B1 - Vector Addition
TutorialTutorial 84B1 - Vector Addition - Answers
MovieBozemann Science - Scalars and Vectors
MovieVideo - Adding Vectors Graphically

 

84B2 - Vector Subtraction

In this section you will learn to:

SUbtract vectors and find the magnitude and direction of the resultant vector.

Subtracting Vectors

Vector subtraction is another vector manipulation most often used in Physics when we need to find the change in a quantity or by how much it has increased or decreased. For example, to calculate acceleration we take the change in velocity and divide by the total time taken. The change in velocity is the final velocity minus the initial velocity and since velocity is a vector, a vector subtraction is needed here.

To subtract vectors we simply add the negative vector that we wish to subtract such that:

A typical vector subtraction to find the change in
velocity.

The resultant vector has magnitude and direction as determined by the vector diagram.

Stuff to Do

TutorialTutorial 84B2 - Vector Subtraction
TutorialTutorial 84B2 - Vector Subtraction - Answers
MovieVideo - Subtracting Vectors Graphically

 


84B3 - Relative Velocity

Here you will learn to:

Calculate the velocity of one object realtive to another using vector subtraction.

Relative Velocity

If two things are moving in a straight line but are travelling at different speeds, then we can work out their relative velocities by simple addition or subtraction as appropriate. For example, imagine two cars travelling along a straight road at different speeds.

If one car (travelling at 30 ms-1) overtakes the other car (travelling at 25 ms-1), then according to the driver of the slow car, the relative velocity of the fast car is +5 ms-1.

In technical terms what we are doing is moving from one frame of reference into another. The velocities of  25 ms-1 and 30 ms-1 were measured according to a stationary observer on the side of the road. We moved from this frame of reference into the driver's frame of reference.

At the beginning of this footage one train is at rest and the other is in
motion and by the end of the footage they reverse roles. This is an
example of relative motion.

Another way to work this out is do a simple vector subtraction:

In words, the velocity of A relative to B is the velocity of A minus the velocity of B or vectorially the velocity of A plus the negative vector for the velocity of B.

Stuff to Do

TutorialTutorial 84B3 - Relative Velocity
TutorialTutorial 84B3 - Relative Velocity - Answers

 

84B4 - Vector Resolution

In this section you will learn to:

Resolve vectors into horizontal and vertical components.

Vector Resolution

The components of a vector are those vectors which, when added together, give the original vector. Working in reverse it is also possible to take a two-dimensional vector and resolve it into two one-dimensional vectors. For example, the vector 5 ms-1 north east can be resoved into two components, 3.5 ms-1 north and 3.5 ms-1 east.

The direction of vectors is always defined relative to a system of axes. For example, in discussing displacement or velocity on the surface of the earth, it is convenient to use axes directed from south to north and from west to east. In such a situation, a displacement A can be thought of as being made up of two components A1 and

A diagram showing how to resolve a vector into its horizontal
and vertical components.

A2 directed along these axes, such that vectorially A = A1 + A2. The components could be determined by constructing a scale diagram or using trigonometry. A1, the component in an easterly direction, will have a magnitude A1 = Acosθ. A2, the component in a northerly direction, will have a magnitude A2 = Asinθ. In all vector problems a natural system of axes presents itself. In many cases the axes are at right angles to one another. Components parallel to the axes of a rectangular system of axes are called rectangular components. In general it is convenient to call the horizontal x-axis and the vertical y-axis. The direction of a vector is given as an angle counter-clockwise from the x-axis.

Stuff to Do

TutorialTutorial 84B4 - Vector Resolution
TutorialTutorial 84B4 - Vector Resolution - Answers