# Resolution of vector into components

## 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.

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