9.0 Resolving vector into its components

  Basic Vectors

Let $\overrightarrow {OP} $ be a position vector of point $P$ in cartesian co-ordinate system.

$\left| {\overrightarrow {OP} } \right| = r:$ Distance of point $P$ from origin or magnitude of the vector $\overrightarrow {OP} $

Vector $\overrightarrow {OP} $ makes an angle of $\theta$ with the $x$ axis as shown in the figure.

In $\Delta ONP$,
$$\cos \theta = \frac{{ON}}{{OP}}$$$$ON = OP\cos \theta $$ or $$ON = r\cos \theta $$ Similarly, $$\sin \theta = \frac{{NP}}{{OP}}$$$$NP = OP\sin \theta $$ or $$NP = r\sin \theta $$ As we know vector is written as, $$\overrightarrow {ON} = \left| {\overrightarrow {ON} } \right|\widehat {ON}$$$$\overrightarrow {ON} = r\cos \theta \widehat i$$ Similarly, $$\overrightarrow {NP} = \left| {\overrightarrow {NP} } \right|\widehat {NP}$$$$\overrightarrow {NP} = r\sin \theta \widehat j$$

In $\Delta ONP$ from vector law of addition,

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$$\overrightarrow {OP} = \overrightarrow {ON} + \overrightarrow {NP} $$$$\overrightarrow {OP} = r\cos \theta \widehat i + r\sin \theta \widehat j$$$$\overrightarrow {OP} = r\left( {\cos \theta \widehat i + \sin \theta \widehat j} \right)$$