Resolution of Vectors Physics Notes

Resolution of Vectors Physics Notes

Resolution of Vectors:
The process of vector addition by the triangle or parallelogram or polygon law is called composition of vectors.
The resolution of a vector into components vector is just the converse of the composition of vectors.
The process of spilliting up a vector into two or more vectors is known as resolution of a vector. The vectors into which a given vector is splitted are called the component vectors.

The component vectors in a given direction gives the measure of the effect of the vector in that direction. p
A given vector may be resolved into any number of component vectors. However we shall study the resolution of a vector into two and three-component vectors.

Resolution of vectors in Two

1. Resolution of a vector In two Dimensions: The resolution of a vector into two mutually perpendicular vectors is called the rectangular resolution of vector in a plane or two dimensions.

Consider that a vector \(\vec{OP}\)= \(\vec{A}\) has to be resolved into two component vector along the direction of two mutually perpendicular directions of X-axis and Y-axis. Let î and ĵ be the unit vectors along X-axis and Y-axis respectively figure

From point P, drop PM and PN perpendicular to X-axis and Y-axis respectively. From the parallelogram law of vector addition, it follows that

The equation (i) describes the rectangular resolution of the vector \(\vec{A}\) into the component vector A_x î and A_y ĵ. In practice Aî and Aĵ are called respectively x-component and y-component of vector A.

Further A_x, and A_y are called the magnitude of the two-component vectors.

If A is the magnitude of the vector \(\vec{A}\) and O is its inclination with X-axis, then from the right angled triangle OMP,

Adding the squares of Ax and Ay, we get

2. Resolution of a vector in Three Dimensions:
Consider that the vector \(\vec{OP}\)= \(\vec{A}\) represents a vector in space. In order to express it in the form of three mutually perpendicular components, construct a rectangular parallel OTBCDEP with three edges along the three co-ordinate axes OX, OY and OZ. Let î, ĵ, and k̂ be the unit vectors along OX, OY and OZ respectively as shown in the figure.

Then according to the polygon law of addition of vectors, we have
\(\overrightarrow{O P}=\overrightarrow{O T}+\overrightarrow{T B}+\overrightarrow{B P}\) ……(i)
If the three sides of the rectangular figure respectively are OT = A_x, OC = A_y and OE = A_z
Then, \(\vec{OT}\) = A_x î, \(\vec{OC}\) = A_y ĵ and \(\vec{OE}\) = A_z k̂

Since, \(\overrightarrow{T B}=\overrightarrow{O C}\) = A_y ĵ and
\(\overrightarrow{B P}=\overrightarrow{O E}=\overrightarrow{C D}\) = A_z k̂

Then the equation (i) becomes
\(\vec{A}\) = A_x î + A_y ĵ + A_z k̂ ……….(ii)

The equation (ii) expresses the vector \(\vec{A}\) oriented in space (three dimensions) in terms of its three rectangular components A_x î, A_y ĵ and A_z k̂.

Magnitude of \(\vec{A}\):
In triangle OBP
OP² = OB² + BP²

In triangle OCB
OB² = OC² + CB² = OC² + OT²
∴ OP² = OC² + OT² + BP²
A² = A_y² + A_x² + A_z²
A² = \(\sqrt{A_{x}^{2}+A_{y}^{2}+A_{z}^{2}}\) …….(iii)
Thus the magnitude of a vector is equal to the square root of the sum of the squares of the magnitude of its rectangular components.

Directions: Cosines of a Vector:
If oc,P and y are the angles which \(\vec{A}\) makes with X, Y and Z axes respectively, then
cos α = \(\frac{A_{x}}{A}\) or A_x = A cos α
cos β = \(\frac{A_{y}}{A}\) or A_y = A cos β
cos γ = \(\frac{A_{z}}{A}\) or A_z = A cos γ

Here cos α, cos β and cos γ are called the direction cosines of the vector \(\vec{A}\)
Putting the value of A_x, A_y and A_z in eq. (iii), we get,
A² = A² cos² α + A² cos² β + A² cos² γ
A² = A² (cos² α + cos² β + cos² γ )
It means that the sum of the squares of the direction cosines of a vector is always unity (1).

Physics Notes