Let A and B be two vectors which are represented by two directed lines OM and ON respectively. The vector B is added to A by the head to tail rule of vector addition (fig 2.9). thus the resultant vector R = A + B is give, in direction and magnitude, by the vector OP.

In the Fig 2.9 ,  vector_op_equationand  are the x components of the vectors A, B and R and their magnitudes are given by the lines OQ, MS, and OR respectively, but

vector_op_equation_2

Which means that the sum of the magnitudes of x-components of two vectors which are to be added, is equal to the x-component of the resultant. Similarly the sum of the magnitudes of y-components of two vectors is equal to the magnitude of y-component of the resultant, that is

y-component_of_the_resultant

Since rectangular_components and  are the rectangular components of the resultant vector R, hence

rectangular_components_1

The magnitude of the resultant vector R is thus given as

rectangular_components_2

And the direction of the resultant vector is determined from

rectangular_components_3

 

Similarly for any number of coplanar vectors A, B, C…….. we can write

rectangular_components_4

Do you know?

The_Chinese_acrobatsThe Chinese acrobats in this incredible balancing act are in equilibrium.

The vector addition by rectangular components consists of the following steps.

quadrant

Table 2.1

i)                    Find x and y components of all given vectors.

ii)                   Find x-component x_and_y_components  of the resultant vector by adding the x-component of all the vectors.

iii)                 Find y-componenty-component of the resultant vector by adding the y-components of all the vectors.

iv)                 Find the magnitude of resultant vector R using

magnitude_of_resultant_vectorv)                  Find the direction of resultant vector R by using

direction_of_resultant_vectorwhere  is the angle, which the resultant vector makes with positive x-axis. The signs of x_and_y_components and y-component determine the quadrant in which resultant vector lies. For that purpose proceed as given below.

Irrespective of the sign of  R and Y_equation , determine the value of determine_the_value from the calculator or by consulting trigonometric tables. Knowing the value of ф, angle θ is determined as follows.

a)      If both  R and Y_equation  are positive, then the resultant lies in the first quadrant and its direction is θ = ф.

b)      If  is x_and_y_components– ive and y-component is +ive, the resultant lies in the second quadrant and its direction is θ = 180⁰ – ф.

c)       If both  R and Y_equation are –ive, the resultant lies in the third quadrant and its direction is θ = 180⁰ – ф.

d)      If x_and_y_components is positive and y-component is negative, the resultant lies in the fourth quadrant and its direction is θ = 360⁰ – ф.

quadrant_graph

Example 2.2: two forces of magnitude 10 N and 20 N act on a body in direction making angles 30⁰ and 60⁰ respectively with x-axis. Find the resultant force.

Solution:

Step (i) x-components

The x-component of the first force : first_forcecos 30⁰

= 10 N x 0.866 = 8.66 N

The x-components of second force = second_force

= 20 N x 0.866 = 17.32 N

Step (ii)

The magnitude of y component fx of the resultant force F

resultant_force_F Step (iii)

The magnitude of y component Fy of the resultant force F

magnitude_of_y Step (iv)

The magnitude F of the resultant force F

the_resultant_force F

Step (v)

F the resultant force F makes and angle θ with the x-axis then

makes_and_angle_θ

Example: 2.3: find the angle between two forces of equal magnitude when the magnitude of their resultant is also equal to the magnitude of either of these forces.

Solution: let θ be the angle between two forces F1 is along x-axis. Then x-component of their resultant will be

angle_between_two_forces And y-component of their resultant is

y-component_of_their_resultant

resultant_R_is_given

Point to ponder

Why do you keep your legs far apart when you have to stand in the aisle of a bumpy-riding bus?