** Resolution of vector into
components**

The process of splitting a vector into various components is called as Resolution of Vector. These parts of a vector may act in different directions and called as components of vector. We can resolve a vector into a number of components. Generally there are three components of vectors. Component along X – axis is called X – component, component along Y – axis is called Y – component and component along Z – axis is called Z – component.

Consider a vector \(\overrightarrow{R}\) in X – Y plane as shown in below figure. If we draw orthogonal vectors \(\overrightarrow{{{R}_{x}}}\) and \(\overrightarrow{{{R}_{y}}}\) along \(x\) and \(y\) axis respectively, by law of vector addition: \(\overrightarrow{R}=\overrightarrow{{{R}_{x}}}+\overrightarrow{{{R}_{y}}}\)

Now as for any vector, \(\overrightarrow{A}=A\widehat{n}\) so, \(\overrightarrow{{{R}_{x}}}=\widehat{i}{{R}_{x}}\) and \(\overrightarrow{{{R}_{y}}}=\widehat{j}{{R}_{y}}\) so,

\(\overrightarrow{R}=\widehat{i}{{R}_{x}}+\widehat{j}{{R}_{y}}\)……….. (1) But from figure \({{R}_{x}}=R\cos \theta \)…… (ii) And \({{R}_{y}}=R\sin \theta \)…..(iii)

Since R and \(\theta \) are usually known, equation (ii) and (iii) give the magnitude of the components of \(\overrightarrow{R}\) along x and y – axis respectively.

Here it is worthy to note once a vector is resolved into its components, the components themselves can be used to specify the vector as:

i) The magnitude of the vector \(\overrightarrow{R}\) is obtained by squaring and adding equation (ii) and (iii), i.e.

\(R=\sqrt{R_{x}^{2}+R_{y}^{2}}\)ii) The direction of the vector \(\overrightarrow{R}\) is obtained by dividing equation (iii) by (ii) i.e.

\(\tan \theta =\left( \frac{{{R}_{x}}}{{{R}_{y}}} \right)\) \(\Rightarrow \theta ={{\tan }^{-1}}\left( \frac{{{R}_{y}}}{{{R}_{x}}} \right)\)