Explain the analytical method for vector addition.
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 $\overrightarrow{\mathrm{B}}$
Similar Questions
Match List$- I$ with List$- II.$
$[Image]$
Choose the correct answer from the options given below :
Match List$- I$ with List$- II.$
$[Image]$
Choose the correct answer from the options given below :
- [JEE MAIN 2021]
Statement $I:$ If three forces $\vec{F}_{1}, \vec{F}_{2}$ and $\vec{F}_{3}$ are represented by three sides of a triangle and $\overrightarrow{{F}}_{1}+\overrightarrow{{F}}_{2}=-\overrightarrow{{F}}_{3}$, then these three forces are concurrent forces and satisfy the condition for equilibrium.
Statement $II:$ A triangle made up of three forces $\overrightarrow{{F}}_{1}, \overrightarrow{{F}}_{2}$ and $\overrightarrow{{F}}_{3}$ as its sides taken in the same order, satisfy the condition for translatory equilibrium.
In the light of the above statements, choose the most appropriate answer from the options given below:
Statement $I:$ If three forces $\vec{F}_{1}, \vec{F}_{2}$ and $\vec{F}_{3}$ are represented by three sides of a triangle and $\overrightarrow{{F}}_{1}+\overrightarrow{{F}}_{2}=-\overrightarrow{{F}}_{3}$, then these three forces are concurrent forces and satisfy the condition for equilibrium.
Statement $II:$ A triangle made up of three forces $\overrightarrow{{F}}_{1}, \overrightarrow{{F}}_{2}$ and $\overrightarrow{{F}}_{3}$ as its sides taken in the same order, satisfy the condition for translatory equilibrium.
In the light of the above statements, choose the most appropriate answer from the options given below:
- [JEE MAIN 2021]
If $a$ and $b$ are two units vectors inclined at an angle of $60^{\circ}$ to each other, then
Which of the following quantity/quantities are dependent on the choice of orientation of the co-ordinate axes?
$(a)$ $\vec{a}+\vec{b}$
$(b)$ $3 a_x+2 b_y$
$(c)$ $(\vec{a}+\vec{b}-\vec{c})$
Which of the following quantity/quantities are dependent on the choice of orientation of the co-ordinate axes?
$(a)$ $\vec{a}+\vec{b}$
$(b)$ $3 a_x+2 b_y$
$(c)$ $(\vec{a}+\vec{b}-\vec{c})$
Two vectors $\overrightarrow{{X}}$ and $\overrightarrow{{Y}}$ have equal magnitude. The magnitude of $(\overrightarrow{{X}}-\overrightarrow{{Y}})$ is ${n}$ times the magnitude of $(\overrightarrow{{X}}+\overrightarrow{{Y}})$. The angle between $\overrightarrow{{X}}$ and $\overrightarrow{{Y}}$ is -
Two vectors $\overrightarrow{{X}}$ and $\overrightarrow{{Y}}$ have equal magnitude. The magnitude of $(\overrightarrow{{X}}-\overrightarrow{{Y}})$ is ${n}$ times the magnitude of $(\overrightarrow{{X}}+\overrightarrow{{Y}})$. The angle between $\overrightarrow{{X}}$ and $\overrightarrow{{Y}}$ is -
- [JEE MAIN 2021]