Composition of Several Vectors by Resolution : Process, Calculation and Solved Examples

Physics Topics can be challenging to grasp, but the rewards for understanding them are immense.

What is the Resolution of Vectors into Two Components?

Three or more vectors can be added by the law of polygon of vectors. This method is geometrically convenient but is inconvenient mathematically. However, when vectors lie on a plane, an elegant method is to resolve each of the vectors into two mutually perpendicular components while performing addition.

Process: The vectors are drawn on a plane from a common initial point. Two mutually perpendicular axes X and Y are drawn from that point and each vector is resolved into components along these two axes. Next, algebraic sums of the x -components and of the y -components of the vectors with proper sign (+ or -) are determined separately. Now, vector addition of the resultant x and y components gives the resultant of all the vectors.

Calculation: Let \(\vec{P}\), \(\vec{Q}\), \(\vec{R}\) be three coplanar vectors having a common initial point O. To find the resultant, axes OX and OY are drawn, coplanar with the vectors [Fig.].

Let \(\vec{P}\), \(\vec{Q}\), \(\vec{R}\) make angles α, β, γ respectively with the x axis. Thus their respective components along x -axis are
P_x = P cosα, Q_x = Q cos β and R_x = R cosγ. Similarly P_y = Psinα, Q_y = Q sin β and R_x = Rsinγ.

Hence, the resultants (sums) F_x and F_y, along x and y axes are
F_x = P cosα + Q cosβ + R cosγ
and F_y = P sinα + Q sinβ + R sinγ

For performing the summation, positive or negative values of sine and cosine functions for the angles α, β, γ should be taken into consideration. In Fig., \(\vec{F}\) represents the resultant of F_x and F_y, according to the triangle or the parallelogram law of vector addition.

F = \(\sqrt{F_x^2+F_y^2}\) ……. (3)
and the inclination O of? with the x -axis is given by
tanθ = \(\frac{F_y}{F_x}\) …… (4)
Equations (3) and (4) give the magnitude and direction respectively of the resultant of the three vectors \(\vec{P}\), \(\vec{Q}\), \(\vec{R}\). The resultant of any number of coplanar vectors can be found out using this method.

Numerical Examples

Example 1.
Compute the resultant of three coplanar vectors P, 2P and 3P inclined at 120° with one another.
Solution:
Let the x-axis be taken along vector P. Then the vectors 2P and 3P will [Fig.] make angles 120° and 240° respectively with the x -axis. Resolving the vectors into components along x and y axes and adding, we get the sums R_x and R_y respectively along x and y axes.

Hence, the value of the resultant R is,
R = \(\sqrt{R_x^2+R_y^2}\) = \(\sqrt{\frac{9}{4} P^2+\frac{3}{4} P^2}\) = \(\sqrt{3}\)P
\(\vec{R}\) makes an angle θ with the x-axis such that,
tanθ = \(\frac{R_y}{R_x}\) = \(\frac{-\frac{\sqrt{3}}{2} P}{-\frac{3}{2} P}\) = \(\frac{1}{\sqrt{3}}\) = tan 30° or, tan 210°
As R_x and R_y), are both negative, R must be in the third quadrant and θ = 210°.

Example 2.
Five coplanar forces, each of magnitude F, are acting on a particle. Each force is inclined at an angle 30° with the previous one. Find out the magnitude and direction of the resultant force on the particle.
Solution:
Let us choose x-axis along the first force, and y-axis along the perpendicular direction on the plane of the forces [Fig.].
The angles made by the five forces with the x -axis are 0°, 30°, 60°, 90° and 120°.

∴ Component of resultant force along x-axis,
F_x = Fcos0° + Fcos30° + Fcos60° + Fcos90° + Fcos120°
= F[1 + \(\frac{\sqrt{3}}{2}\) + \(\frac{1}{2}\) + 0 – \(\frac{1}{2}\)]
= F(1 + \(\frac{\sqrt{3}}{2}\))
and component of resultant force along y-axis,

The angle of inclination of F’ with the x-axis,
θ = tan^-1\(\frac{F_y}{F_x}\) = tan^-1\(\sqrt{3}\) = 60° (along the third force)

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