What is Vector quantity? Vector quantities are physical quantities that have both magnitudes and directions. Examples of vector quantities are displacement, velocity, force, displacement and many more.
What is scalar quantity? Scalar quantities are physical quantities that have magnitudes, but no directions. Examples are, distance, speed, mass, temperature, etc, .
Scalar quantities unlike vector quantities can be added and subtracted using ordinary algebra.
A vector can be represented by line OA, as show in the diagram below. The magnitude of vector is represented by the length of the line OA. If a vector is refers to be OA, it means that the vector have a magnitude equivalent to the length of OA, and act from O to A
A vector of 40 units can be represented on paper by an equivalent length, for example let say, 5cm, it means that 1cm, represent 8units.
The direction of the vector is represented by angle which it
makes with a give line OX, as shown on the diagram, by the arrow.
The parallelogram law of vector, states that if two vectors are represented in magnitudes and directions, by the adjacent sides of a parallelogram, the resultant is represented in magnitude and direction by the diagonal of the parallelogram, drawn from the common points.
Two or more vectors acting on a body in given direction can
be added or combined to give a single vector, which produce the same effect.
The single vector produced after combining two or more vectors is called the resultant.
Consider two forces of magnitude P=30N and Q=40N, acting
on a body O in any of the following
In this case the resultant R, of the forces, P and Q , is given by the algebraic sum of the two force, that is
R= P + Q = 30 + 40 + 70N along OX
(B).P and Q act in opposite direction
For this number (B), the resultant of the two forces is
obtained by subtraction, since they are acting in opposite direction to each
other; therefore the resultant is given as
R = Q – P = 40 – 30 = 10N along OX
(C).P and Q act at right angle to each other
In the case where the forces are inclined to each other at an angle, the resultant of the two forces cannot be obtained by simple algebraic subtraction, or addition, but by what is called vector addition.
To get the resultant of the vectors inclined to each other,
we can use the parallelogram law of vector,
trigonometry ratio, or scalar drawing to obtain the magnitude and
directions of the resultant vector.
So from the
Pythagoras’s theorem the solution of (c), is given as;
R2 = P2 + Q2
The direction of the resultant with respect to a given line OX is the angle which R makes with OQ. It is given by:
The HTML Clipboard
= , assuming that P=3 and Q =4, then
In alternative method, the resultant R and the direction α of the vector can also be found using scale drawing. The P and Q are represented using any suitable scale , for example 1cm to represent 10N, so that P (30N), is represented by 3cm and Q (40N) is represented by 4cm. The rectangle is completed as shown below.
Now you, measure, the diagonal and convert it into Newton, using
the same scale of 1cm=10N to obtain
the resultant R.
The direction of the resultant
vector, which is the angle α, can
also be measured using a protractor.
If you have done your scale drawing well, you will get
approximately the same value like the one you got using Pythagoras theorem.
(D).P and Q act at an acute angle Ѳ =60o
If p and Q are inclined at angle 60o, . In this case, you can use cosine rule to obtain
the resultant that is using
R2 =P2 + Q2 – 2PQcos (180-Ѳ)
And because cos (180-Ѳ) = – cosѲ, then we have
R2 = P2 + Q2 + 2PQcosѲ
Where Ѳ is the
angle between P and Q,
And α, can be found using the sine rule
But sine 120o = sin (180o -120o) = sin60o, then we have
You can also use scale drawing to obtain, the resultant and the direction, by using scale of 1cm = 10N, as shown, below. Again we use scale of 1cm=10N. we draw the length 4cm to represent Q ( 40N), using protractor , we draw the angle 60o, then we draw length 3cm to represent P (30N). Then we complete the parallelogram, as shown with dotted lines. The resultant and its direction can now be obtained by measuring the resultant R, with meter and measuring the direction of it which is the angle α between OA, and OQ using protractor
(E).P and Q act at an obtuse angle O-120o
If P and Q are inclined to at 120o,
the resultant can also be solved using cosine
rule, that is
A vector, such as force OA =10N, acting at angle 30o, with direction of OX, can be broken or resolved into two perpendicular parts. This component or parts can be determined by drawing a rectangle for which the force will be represented by diagonal
Using Pythagoras’s theorem, we can obtain
OX =10cos30o (horizontal component) and
OY=10sin30o (vertical component)
These two components called horizontal and vertical
component can also be resolved to produce original result, by using vector addition
To find the resultant
of several forces, f1,
f2,f3 f4…… acting on a point O, we
first reduce the whole system to two perpendicular forces and then compound the
force to obtain the resultant
To reduce the system to two perpendicular forces, we simple find the component of each forces, f1, f2,f3,…along two perpendicular direction OX and OY, and then add to obtain P and Q
Let look at the example below, Find the resultant of the system of forces given in diagram below
Stage A, is the given problem needed to be solved, so from stage A, it was reduced to stage B, and to stage C, where it is finally reduced to two perpendicular force of 4N and 2N. From stage C, the vectors can now be solved using Pythagoras theorem.
Where R = resultant
, the direction
is given by tan-12=63o with horizontal