# Resolution of Vectors and Rectangular Components

## Resolution of Vectors and Rectangular Components

The process of a splitting a vector is called resolution of a vector. In simpler language it would mean, determining the effect of a vector in a particular direction. The parts of the vector obtained after splitting the vector are known as Components of the Vector.

Rectangular components of a vector: If the components of a given vector are perpendicular to each other, they are called as Rectangular components. The figure illustrates a vector $$\overrightarrow{A}$$ represented by $$\overrightarrow{OP}$$. Through the point, O two mutually perpendicular axis X and Y are drawn. From the point P, two perpendicular, PN and PM are dropped on X and Y axis respectively. The vector $$\overrightarrow{{{A}_{x}}}$$ is the resolved part of $$\overrightarrow{A}$$ along the X – axis. It also known as the X – component of $$\overrightarrow{A}$$ and is the projection of the $$\overrightarrow{A}$$ on X- axis. Similarly, $$\overrightarrow{{{A}_{y}}}$$ is the resolved part of the $$\overrightarrow{A}$$ along the Y – axis, and is therefore, known as the Y – component of $$\overrightarrow{A}$$.

Applying the law of triangle of vectors to ONP,  $$\overrightarrow{OP}\,\,=\,\,\overrightarrow{ON}\,+\,\overrightarrow{NP}$$ or $$\overrightarrow{A}\,=\,\overrightarrow{{{A}_{x}}}\,\,+\,\,\overrightarrow{{{A}_{y}}}$$, which also confirm that Ax, Ay are the components of A.

Moreover, in the right – angled MONP,

$$\cos \,\theta \,\,=\,\,\frac{{{A}_{x}}}{A}$$

⇒ Ax = A cosθ … (1) $$\sin \,\theta \,=\,\frac{{{A}_{y}}}{A}$$

⇒ Ax = A sinθ … (2)

Squaring and adding equations (1) and (2) we get,

Ax² + Ay² = A² cos²θ + A² sin²θ = A² (cos²θ + sin²θ)

But, cos²θ + sin²θ = 1

∴ Ax² + Ay² = A²

⇒ A² = Ax² + Ay²

$$A\,=\,\sqrt{{{A}_{x}}^{2}\,\,+\,\,{{A}_{y}}^{2}}$$

This equation gives the magnitude of the given vector in terms of the magnitudes of the components of the given vector.

In the figure, the velocity vector $$\overrightarrow{V}$$ is represented by the vector $$\overrightarrow{OP}$$. Resolving $$\overrightarrow{V}$$ into its two rectangular components, we have $$\overrightarrow{V}\,=\,\overrightarrow{{{V}_{x}}}+\overrightarrow{{{V}_{y}}}$$. In terms of the unit vectors $$\widehat{i}$$, $$\widehat{j}$$, $$\overrightarrow{V}\,\,=\,{{V}_{x}}\,\widehat{i}\,\,+\,{{V}_{y}}\widehat{j}$$

Where,

Vx = V cosθ, Vy = V sinθ and $$\tan \,\theta \,=\,\frac{{{V}_{y}}}{{{V}_{x}}}$$.

How to find the Resolution of Vectors?

Problem: Diagram above shows two forces of magnitude 25N are acting on an object of mass 2kg. Find the acceleration of object, in m/s². Solution: Given,

Mass (m) = 2kg

Horizontal component of the forces = 25 cos 45° + 25 cos 45° = 35.36 N

Vertical component of the forces = 25 cos 45° – 25 cos 45° = 0 N

The acceleration of the object can be determined by the equation Force (F) = mass (m) x acceleration (a)

(35.36) = 2 x a

Acceleration (a) = 35.36/3 = 17.68 m/sec²