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
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
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
Step 3 : Choose a positive direction
Lets make to the right the positive direction. This means that to the left becomes the
Step 4 : Now define our vectors algebraically
With right positive:
Step 5 : Add the v
= 8 sin 135
Thus, F A C = 62.8o .
Step 4 : Quote the resultant
Our final answer is then:
Resultant Displacement: 18.5km on a bearing of 62.8o
Components of Vectors
In the discussion of vecto
Figure 4.1: Illustration of Displacement
Definition: Displacement is a vector with direction
pointing from some initial (starting) point to some final (end) point and whose
magnitude is the straight-line distance from the st
Step 2 : Choose a suitable scale
In this problem a scale of 1cm = 0.5N would be appropriate, since then the vector diagram
would take up a reasonable fraction of the page. We can now begin the accurate scale
Step 3 : Draw the first scaled vector
Step 4 : Now define our vectors algebraically
With right positive:
v f inal
Step 5 : Subtract the vectors
Thus, the change in velocity of the ball is:
= (2m.s1 ) (+3m.s1 )
Step 6 : Quote the resultant
Step 2 : Determine the length of the resultant
Note that the triangle formed by his separate displacement vectors and his resultant
displacement vector is a right-angle triangle.
We can thus use Pythogoras theorem to
determine the length of th
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
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
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
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
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
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
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
The Parallelogram Method
When needing to find the resultant of two vectors another graphical technique can be applied- the
parallelogram method. The following strategy is employed:
Choose a scale and a reference direction.
Choose either of the vectors t
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