TOPIC 6: VECTORS ~ MATHEMATICS FORM 4
If an object moves from point A to another point say B, there is a displacement
There
are many Vector quantities, some of which are: displacement, velocity,
acceleration, force, momentum, electric field and magnetic field.
Normally vectors are named by either two capital letters with an arrow above e.g.
In
the x —plane all vectors with initial points at the origin and their
end points elsewhere are called position vectors. Position vectors are
named by the coordinates of their end points.
Magnitude and Direction of a Vector
The magnitude / modules of a vector is the size of a vector, it is a
scalar quantity that expresses the size of a vector regardless of its
direction.
Normally the magnitude of a given vector is calculated by using the distance formula which is based on Pythagoras theorem.
The direction of a Vector may be given by using either bearings or direction Cosines.
Reading bearings: There
are two method used to read bearings, in the first method all angles
are measured with reference from the North direction only where by the
North is taken as 0000, the east 0900, the South is 1800 and West 2700
From the figure above, point P is located at a bearing of 0500, while Q is located at a bearing of 1350.
Commonly
the bearing of point B from point A is measured from the north
direction at point A to the line joining AB and that of A from B is
measured from the North direction at point B to the line joining BA.
From the figure above the bearing of B from A, is 0600 while that of A from B is 2400 In the second method two directions are used as reference directions, these are North and south.
In
this method the location of places is found by reading an acute angle
from the north eastwards or westwards and from the south eastwards or
westwards.
From figure above, the direction of point A from O is N 460 E , that of B is N500W while the direction from of C is S200E.
Example 7
Mikumi is 140km at a bearing of 0700f from Iringa. Makambako is 160km at bearing of 2150
from Iringa. Sketch the position of these towns relative to each other,
hence calculate the magnitude and direction of the displacement from
Makambako to Mikumi.
Alternatively
by using the scale AB is approximately14.3 cm Therefore AB = 14.3x 20
km = 286km and the bearing is obtained a protractor which is about N510E
The Sum of Two or More Vectors
Find the sum of two or more vectors
Adding
two vectors involves joining two vectors such that the initial point of
the second vector is the end point of first vector and the resultant is
obtained by completing the triangle with the vector whose initial point
is the initial point of the first vector and whose end points the end
point of the second vector.
(2) The parallelogram law
two vectors have a common initial point say P, then their resultant is
obtained by completing a parallelogram, where the two vectors are the
sides of the diagonal through P and with initial point at P
Example 9
Note that by parallelogram law of vector addition, commutative property is verified.
you want to add more than two vectors, you join the end point to the
initial point of the vectors one after another and the resultant is the
vector joining the initial point of the first vector to the end point of
the last vector
Example 10
In the figure above P is the initial point of a, b has been joined toaat point Q and c is joined to b at R, while d is joined to c at point S and PT = a + b + c + d which is the resultant of the four vectors.
Two vectors are said to be opposite to each other if they have the same magnitude but different directions
The Difference of Vectors
Find the difference of vectors
when subtracting one vector from another the result obtained is the
same as that of addition but to the opposite of the other vector.
Consider the following figure
A Vector by a Scalar
Multiply a vector by a scalar
If
a vector U has a magnitude m units and makes an angleθwith a positive x
axis, then doubling the magnitude of U gives a vector with magnitude
2m.
Vectors in Solving Simple Problems on Velocities, Displacements and Forces
Apply vectors in solving simple problems on velocities, displacements and forces
from the dormitory to the parade ground and then he walks 100m due east
to his classroom. Find his displacement from dormitory to the
classroom.
Consider the following figure describing the displacement which joins the dormitory D. parade ground P and Classroom C.
- Determine the magnitude and direction of their resultant.
- Calculate the magnitude and direction of the opposite of the resultant force.
Fo= 13N and its bearing is (67.40+1800) = 247.40
So the magnitude and direction of the force opposite to the resultant force is 13N and S67.40W respectively..
Exercise 2
- The resultant of U + V + W
- The magnitude and direction of the resultant calculated in part (a) above.
A boat moves with a velocity of 10km/h upstream against a downstream
current of 10km/h. Calculate the velocity of the boat when moving down
steam.
Calculate the magnitude
and direction of the resultant of the velocities V1=5i + 9j,V2 = 4i + 6j and V3 = 4i – 3j where i and j are unit vectors of magnitude 1m/s in the positive directions of the x and y axis respectively.
