Question
Prove that: Vector addition is commutative.
✓
Answer
Commutative property of vector addition:
According to commutative property, for two vectors $\vec{P}$ and $\vec{Q}, \vec{P}+\vec{Q}=\vec{Q}+\vec{p}$
Proof:
i. Let two vectors $\overrightarrow{ P }$ and $\overrightarrow{ Q }$ be represented in magnitude and direction by two sides O$\overrightarrow{ OA }$ and $\overrightarrow{ AB }$ respectively.
ii. Complete a parallelogramOABC such that $\overrightarrow{ OA }=\overrightarrow{ CB }=\overrightarrow{ P }$ and $\overrightarrow{ AB }=\overrightarrow{ OC }=\overrightarrow{ Q }$ then join OB
iii. $\ln \triangle OAB , \overrightarrow{ OA }+\overrightarrow{ AB }=\overrightarrow{ OB }$
(By triangle law of vector addition)
$\therefore \vec{P}+\vec{Q}=\vec{R}$
In $\triangle O C B, \overrightarrow{O C}+\overrightarrow{C B}=\overrightarrow{O B}$
(By triangle law of vector addition)
$\therefore \vec{Q}+\vec{P}=\vec{R} \ldots \text { (2) }$
iv. From equation (1) and (2),
$\overrightarrow{ P }+\overrightarrow{ Q }=\overrightarrow{ Q }+\overrightarrow{ P }$
Hence, addition of two vectors obeys commutative law.
According to commutative property, for two vectors $\vec{P}$ and $\vec{Q}, \vec{P}+\vec{Q}=\vec{Q}+\vec{p}$
Proof:
i. Let two vectors $\overrightarrow{ P }$ and $\overrightarrow{ Q }$ be represented in magnitude and direction by two sides O$\overrightarrow{ OA }$ and $\overrightarrow{ AB }$ respectively.
ii. Complete a parallelogramOABC such that $\overrightarrow{ OA }=\overrightarrow{ CB }=\overrightarrow{ P }$ and $\overrightarrow{ AB }=\overrightarrow{ OC }=\overrightarrow{ Q }$ then join OB
iii. $\ln \triangle OAB , \overrightarrow{ OA }+\overrightarrow{ AB }=\overrightarrow{ OB }$
(By triangle law of vector addition)
$\therefore \vec{P}+\vec{Q}=\vec{R}$
In $\triangle O C B, \overrightarrow{O C}+\overrightarrow{C B}=\overrightarrow{O B}$
(By triangle law of vector addition)
$\therefore \vec{Q}+\vec{P}=\vec{R} \ldots \text { (2) }$
iv. From equation (1) and (2),
$\overrightarrow{ P }+\overrightarrow{ Q }=\overrightarrow{ Q }+\overrightarrow{ P }$
Hence, addition of two vectors obeys commutative law.
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