In figure (a), $\vec{A}$ and $\vec{B}$ vectors are coplanar and non-parallel.

$\overrightarrow{\mathrm{R}}$ is to be resolved in $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$

Suppose, $\overrightarrow{O Q}$ represent $\vec{R}$

In figure (b), draw a line parallel to $\vec{A}$ from $O$ and draw another line parallel to $\vec{B}$ passes through Q. Both lines intersect at P.

As per triangle method for vector addition,

$\overrightarrow{\mathrm{OQ}}=\overrightarrow{\mathrm{OP}}+\overrightarrow{\mathrm{PQ}}$

Here, $\overrightarrow{\mathrm{OP}} \| \overrightarrow{\mathrm{A}} \quad \therefore \overrightarrow{\mathrm{OP}}=\lambda \overrightarrow{\mathrm{A}}$

and $\overrightarrow{\mathrm{PQ}} \| \overrightarrow{\mathrm{B}} \quad \therefore \overrightarrow{\mathrm{PQ}}=\mu \overrightarrow{\mathrm{B}}$

(Here, $\lambda$ and $\mu$ are scaler values)

$\therefore \overrightarrow{\mathrm{R}}=\lambda \overrightarrow{\mathrm{A}}+\mu \overrightarrow{\mathrm{B}}$

OR

$\overrightarrow{\mathrm{R}}=($ Component of $\overrightarrow{\mathrm{R}}$ in direction of $\overrightarrow{\mathrm{A}})+($ Component of $\overrightarrow{\mathrm{R}}$ in direction of $\overrightarrow{\mathrm{B}})$