Figure 4.2: An example of two vectors being added to give a resultant
Vector addition using components
In Fig 4.3 two vectors are added in a slightly different way to the methods discussed so far. It might look a
little like we are making more work
tangent can be used as an illustration. Else defer until chapter on Graphs and Equations of Motion.
Instantaneous velocity: reading on the speedometer in a direction tangent to the path. Instantaneous
speed is magnitude of instantaneous velocity but avera
Figure 4.3: Components of vectors can be added as well as the vectors themselves
Step 2 : Resolve the red vector into components
Let us start with the bottom vector. If you are told that this vector has a length of
5.385 units and an angle of 21.8o to the
Note that the angle of incidence i is between the incident ray and the normal 2 to the
not between the incident ray and the surface of the mirror.
There are two forms of an image formed by reflection: real and virtual.
A real image is formed by the actual
and east). Algebraic techniques, however, are not limited to cases where the vectors to be combined
are along the same straight line or at right angles to one another. The following example illustrates
Worked Example 11
Further example of vector add
Image - located at infinity Ob ject
between f and P diag
Image - located behind the mirror - erect (right side up) - magnified (increased in size, larger)
diag base ray diag! define things
Image - located behind the mirror -
Using our known trigonometric ratios we can calculate the value of ;
6 tan =
= 36.8o .
Step 6 : Quote the final answer
Our final answer is a resultant of 10 units at 36.8o
to the positive x-axis.
Do I re
Step 4 : Apply the scale conversion
We now use the scale to convert the length of the resultant in the scale diagram to the actual
displacement in the problem. Since we have chosen a scale of 1cm = 1km in this problem the
resultant has a magnitude of 8.38
In a rough sketch one should include all of the information given in the problem. All of the
magnitudes of the displacements are shown and a compass has been included as a reference
Step 2 : Next we choose a scale for our vector diagram
It is c
This type of microscope uses two convex lenses. The first creates a real magnified image of the object that
is in turn used by the second lens to create the final image. This image is virtual and again enlarged. The
250 cos 30o
= 216.5 km
and sE are the magnitudes of the components they are in the directions north and east respectively.
(NOTE TO SELF: SW: alternatively these results can be arrived at by construction. Include?)
Block on an i
We now know the lengths of the sides of the triangle for which our vector is the hypotenuse.
If you look at these sides we can assign them directions given by the dotted arrows. Then our
original red vector is just the sum of the two dotted vectors (its c
b gives a new vector
c = b
b + (
This clearly shows that subtracting vector
b is the same as adding (
a ) to
In mathematical form, subtracting
Look at the following examples of vector subtraction.
Take the next vector and draw it as an arrow starting from the arrowhead of the first vector in the
correct direction and of the correct length.
Continue until you have drawn each vector each time starting from the head of the previous vector. In
(NOTE TO SELF: This is actually average velocity. For instantaneous s change to dif- ferentials.
Explain that if is large then we have average velocity else for infinitesimal time interval instantaneous!)
What then is speed? Speed is how quickly something
and still function. However since total internal reflection only occurs when light is going from a medium to a
less dense medium, it is necessary ti coat each fibre with glass of a lower refractive index. Otherwise light
would leak from one fibre at their
de-fi4ning -a3ngles -i2, r
Light is refracted according to the laws of refraction:
Laws of Refraction
Laws of Refraction:
sin i /sin r is a constant for two given media (Snells Law)
The incident ray,
Light is at first, something we feel incredibly familiar with. It can make us feel warm, it allows us to see,
allows mirrors and lenses to work, allows for .
Under more careful study light exhibits many fascinating and wonderful propertie
Let us test the first one. It says one step forward and then another step forward is the same as an arrow
twice as long two steps forward.
It is possible that you end up back where you started. In this case the net result of what you have done is
Step 2 : Now we determine speed from the distance and time.
We know that speed is distance covered per unit time. So if we divide the distance
covered by the time it took we will know how much distance was covered for every unit of
The parallelogram method is restricted to the addition of just two vectors. However, it is arguably the
most intuitive way of adding two forces acting at a point.
Algebraic Addition and Subtraction of Vectors
Vectors in a Straight Line
Now you have discovered one use for vectors; describing resultant displacement how far and in what
direction you have travelled after a series of movements.
Although vector addition here has been demonstrated with displacements, all vectors behave i
We know which way the man is running around the track and we know his
speed. His velocity at point B will be his speed (the magnitude of the
velocity) plus his direction of motion (the direction of his velocity). The
instant that he arrives at B he is mov
1cm = 5m
Step 6 : Determine the resultant vector
Now we connect the starting point to the end point and measure the length and direction of
this arrow (the resultant)
1cm = 5m
Step 7 : Apply the scale conversion
Finally we us
(NOTE TO SELF: SW: initially this chapter had a very mathematical approach. I have toned this down
and tried to present in a logical way the techniques of vector manipulation after first exploring the
mathematical properties of vectors.
Mathematical Properties of Vectors
Vectors are mathematical objects and we will use them to describe physics in the
language of mathematics. However, first we need to understand the mathematical
properties of vectors (e.g. how they add and subtract).
Total Internal Reflection
This can only happen when light is travelling into a less dense medium
Refrectionnumbers are n1 = 2 and n2 = 1:
air-7glass -c6ritical5angle-4i r
When light travels from g
= (6)2 + (8)2
The resultant has length of 10 units so all we have to do is calculate its direction. We can specify
the direction as the angle the vectors makes with a known direction. To do this you only need to
visualise the vector
Construction Step 3: Since the ship is now at port B we draw the third vector
5.5cm long starting from this point in the direction south-west. A protractor is re- quired to
measure the angle of 45o .
1cm = 1km
Construction Step 4: