vector and scalars / vector addition and subtraction / vector resolution / relative velocity
Vectors and Scalars?
**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 applicable to natural numbers can manipulate scalar quantities. Many physical quantities can be added together in the same way as natural numbers. For example, if we first put 100 cm^{3} of water into a cup and then put in an additional 150 cm^{3} , the cup will contain 250 cm^{3} 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.
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**
A vector is a quantity that has both a magnitude and a direction. Vectors arise naturally as physical quantities. Examples of vectors are displacement, velocity, acceleration, force and electric field.
Special arithmetic rules must be obeyed when adding vectors together. Much of this topic is devoted to these rules! Examples of vectors include displacement, velocity, force and acceleration.
These physical quantities cannot be added in the simple way 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 7 m! To find the answer, one could draw a scale diagram (1 cm = 1 m) such as is shown on the right. One could also calculate the distance from the starting point using the theorem of Pythagoras, i.e.
It is also useful to know in which direction one has moved from the starting point. This can also be measured from the diagram or calculated from simple trigonometry:
You could have reached the same final position by walking 5 m in the direction 36.9° 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 36.9° east of north require both a magnitude and a direction to fully describe them. These quantities are called displacements. Displacement is an example of a vector quantity.
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 displacements.
**Vector Notation**
Vectors are distinguished from scalars by writing them in special ways. A widely used convention is to denote a vector quantity in bold type, such as A, and that is the convention that will be used. In some books, you may also encounter the letter with an arrow above it.
The magnitude of a vector A is written as |A|.
Vector Representation
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, 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:
- The reference direction is indicated.
- The scale is indicated.
- The vectors are represented as arrows with a length proportional to their magnitude and are correctly orientated with respect to the reference direction.
- 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 above 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.
**Equality of Vectors**
Two vectors are equal when they have the same magnitude and direction, irrespective of their point of origin. In the diagram below, A = B, since they have the same magnitude and direction.
**Negative Vectors**
A vector having the same magnitude but opposite direction to a vector A is -A.
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Vector Addition and Subtraction
Two or more vectors may be added together to produce their addition. If two vectors have the same direction, their resultant has a magnitude equal to the sum of their magnitudes and will also have the same direction.
Similarly orientated vectors can be subtracted the same manner.
It follows that vectors can also be multiplied by a scalar, so for example if the vector A were multiplied by the number m, the magnitude of the vector, |A|, would increase to m|A|, but its direction would not change.
In general, since vectors may have any direction, we must use one of two methods for adding vectors. These are, the polygon method and the parallelogram method.
**Polygon Method**
Two vectors A and B are added by drawing the arrows which represent the vectors in such a way that the initial point of B is on the terminal point of A. The resultant C = A + B, is the vector from the initial point of A to the terminal point of B.
Many vectors can be added together in this way by drawing the successive vectors in a head-to-tail fashion.
If the polygon is closed, the resultant is a vector of zero magnitude and has no direction. This is called the null vector, or 0.
**Parallelogram Method**
In the parallelogram method for vector addition, the vectors are translated, (i.e., moved) to a common origin and the parallelogram constructed as follows:
The resultant R is the diagonal of the parallelogram drawn from the common origin.
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Vector Resolution
The components of a vector are those vectors which, when added together, give the original vector. The sum of the components of two vectors is equal to the sum of these two vectors.
If components are appropriately chosen, this theorem can be used as a convenient method for adding vectors.
The direction of vectors is always defined relative to a system of axes. For example, in discussing displacement 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, an arbitrary displacement A can be thought of as being made up of two components A1 and A2 directed along these axes, such that A = A1 + A2.
The components could be determined by constructing a scale diagram, but they are easily calculated as follows (Note: It is convenient to specify Q clockwise from North when referring to displacements on the earth:
A1, the component in an easterly direction, will have a magnitude |A1| = |A| cosQ.
A2, the component in a northerly direction, will have a magnitude |A2| = |A| sinQ
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 axis X and the vertical axis Y. The direction of a vector is given as an angle counter-clockwise from the X-axis.
The magnitude of A, |A| and the angle can be calculated from the components, using
Note that if a vector is directed along one of the axes, then the component along the other axis is zero.
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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.
Another way to work this out is do a simple vector addition:
In words, the velocity of *a* relative to *b* is the velocity of *a* plus the negative vector for the velocity of *b*.
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download worksheet 8.4.2.A - vectors and relative velocity
download experiment 3 - vectors |