Explain the analytical method for vector addition.

It is much easier to add vectors by combining their respective components.

Consider two vectors $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$ in $x y$-plane with components $\mathrm{A}_{x}, \mathrm{~A}_{y}$ and $\mathrm{B}_{x}, \mathrm{~B}_{y}$

$\therefore \overrightarrow{\mathrm{A}}=\mathrm{A}_{x} \hat{i}+\mathrm{A}_{y} \hat{j}$

$\therefore \overrightarrow{\mathrm{B}}=\mathrm{B}_{x} \hat{i}+\mathrm{B}_{y} \hat{j}$

Let $\overrightarrow{\mathrm{R}}$ be their sum.

$\overrightarrow{\mathrm{R}}=\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}$

$\therefore \overrightarrow{\mathrm{R}}=\left(\mathrm{A}_{x} \hat{i}+\mathrm{A}_{y} \hat{j}\right)+\left(\mathrm{B}_{x} \hat{i}+\mathrm{B}_{y} \hat{j}\right)$

Since vectors obey the commutative and associative laws.

$\therefore \overrightarrow{\mathrm{R}}=\left(\mathrm{A}_{x}+\mathrm{B}_{x}\right) \hat{i}+\left(\mathrm{A}_{y}+\mathrm{B}_{y}\right) \hat{j}$

$\therefore \overrightarrow{\mathrm{R}}=\mathrm{R}_{x} \hat{i}+\mathrm{R}_{y} \hat{j}$

$\mathrm{R}_{x}=\mathrm{A}_{x}+\mathrm{B}_{x}$

$\mathrm{R}_{y}=\mathrm{A}_{y}+\mathrm{B}_{y}$

Thus, each component of the resultant vector $\overrightarrow{\mathrm{R}}$ is the sum of the corresponding components of $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$.

It is much easier to add vectors by combining their respective components.

Consider two vectors $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$ in $x y$-plane with components $\mathrm{A}_{x}, \mathrm{~A}_{y}$ and $\mathrm{B}_{x}, \mathrm{~B}_{y}$

$\therefore \overrightarrow{\mathrm{A}}=\mathrm{A}_{x} \hat{i}+\mathrm{A}_{y} \hat{j}$

$\therefore \overrightarrow{\mathrm{B}}=\mathrm{B}_{x} \hat{i}+\mathrm{B}_{y} \hat{j}$

Let $\overrightarrow{\mathrm{R}}$ be their sum.

$\overrightarrow{\mathrm{R}}=\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}$

$\therefore  \overrightarrow{\mathrm{R}}=\left(\mathrm{A}_{x} \hat{i}+\mathrm{A}_{y} \hat{j}\right)+\left(\mathrm{B}_{x} \hat{i}+\mathrm{B}_{y} \hat{j}\right)$

Since vectors obey the commutative and associative laws.

$\therefore \overrightarrow{\mathrm{R}}=\left(\mathrm{A}_{x}+\mathrm{B}_{x}\right) \hat{i}+\left(\mathrm{A}_{y}+\mathrm{B}_{y}\right) \hat{j}$

$\therefore \overrightarrow{\mathrm{R}}=\mathrm{R}_{x} \hat{i}+\mathrm{R}_{y} \hat{j}$

$\mathrm{R}_{x}=\mathrm{A}_{x}+\mathrm{B}_{x}$

$\mathrm{R}_{y}=\mathrm{A}_{y}+\mathrm{B}_{y}$

Thus, each component of the resultant vector $\overrightarrow{\mathrm{R}}$ is the sum of the corresponding components of $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$