Explain the resolution of a vector in two dimensions. Explain the resolution of a vector into its perpendicular components.

(N/A) Consider an $O$-$XY$ two-dimensional Cartesian coordinate system as shown in the figure.
Let the position vector of point $P$ be $\vec{A}$.
By drawing a projection from $P$ to the $X$-axis,we obtain $OM$,where $\vec{OM} = \vec{A}_x = A_x \hat{i}$ is the $X$-component of $\vec{A}$.
By drawing a projection from $P$ to the $Y$-axis,we obtain $ON$,where $\vec{ON} = \vec{A}_y = A_y \hat{j}$ is the $Y$-component of $\vec{A}$,where $A_x$ and $A_y$ are real numbers.
From the figure,by the parallelogram law of vector addition:
$\vec{A} = \vec{A}_x + \vec{A}_y$
$\vec{A} = A_x \hat{i} + A_y \hat{j}$
Suppose $\vec{A}$ makes an angle $\theta$ with the $X$-axis.
In the right-angled triangle $\Delta OMP$:
$\cos \theta = \frac{A_x}{A} \implies A_x = A \cos \theta$
$\sin \theta = \frac{A_y}{A} \implies A_y = A \sin \theta$
From these equations,it is evident that the components can be positive,negative,or zero depending on the angle $\theta$.
Vectors can be represented in a plane in two ways:
$(i)$ By their magnitude and direction.
$(ii)$ By their components ($X$ and $Y$ components).