Explain the analytical method for vector addition.

(N/A) The analytical method for vector addition involves adding the corresponding components of the vectors.
Consider two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ in the $xy$-plane with components $(A_{x}, A_{y})$ and $(B_{x}, B_{y})$ respectively.
$\overrightarrow{A} = A_{x} \hat{i} + A_{y} \hat{j}$
$\overrightarrow{B} = B_{x} \hat{i} + B_{y} \hat{j}$
Let $\overrightarrow{R}$ be the resultant vector such that $\overrightarrow{R} = \overrightarrow{A} + \overrightarrow{B}$.
Substituting the component forms:
$\overrightarrow{R} = (A_{x} \hat{i} + A_{y} \hat{j}) + (B_{x} \hat{i} + B_{y} \hat{j})$
Since vector addition is commutative and associative,we can group the components:
$\overrightarrow{R} = (A_{x} + B_{x}) \hat{i} + (A_{y} + B_{y}) \hat{j}$
If we write $\overrightarrow{R} = R_{x} \hat{i} + R_{y} \hat{j}$,then by comparing the components,we get:
$R_{x} = A_{x} + B_{x}$
$R_{y} = A_{y} + B_{y}$
Thus,each component of the resultant vector $\overrightarrow{R}$ is the sum of the corresponding components of $\overrightarrow{A}$ and $\overrightarrow{B}$.