Prove the associative law of vector addition.
Prove the associative law of vector addition.
Consider the vectors of shown in the figure. To find the associative law of $\vec{A}, \vec{B}$ and $\vec{C}$ Draw $\vec{A}=\overrightarrow{O P}, \vec{B}=\overrightarrow{P Q}$ and $\vec{C}=\overrightarrow{Q R}$
By triangle law of vector we get, Figure $(b)$
From the figure $\Delta \mathrm{OPQ}$
$\rightarrow \quad \rightarrow \quad \rightarrow \quad \rightarrow$
$\mathrm{A}+\mathrm{B}=\mathrm{OP}+\mathrm{PQ}$
$\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}=\overrightarrow{\mathrm{OQ}}$
By adding $\overrightarrow{\mathrm{C}}=\overrightarrow{\mathrm{QR}}$ in both the sides
$(\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}})+\overrightarrow{\mathrm{C}}=\overrightarrow{\mathrm{OQ}}+\overrightarrow{\mathrm{QR}}$
We get $\quad(\vec{A}+\vec{B})+\vec{C}=\overrightarrow{O R} \ldots \ldots$ $(i)$
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- [AIIMS 2009]
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Establish the following vector inequalities geometrically or otherwise:
$(a)$ $\quad| a + b | \leq| a |+| b |$
$(b)$ $\quad| a + b | \geq| a |-| b |$
$(c)$ $\quad| a - b | \leq| a |+| b |$
$(d)$ $\quad| a - b | \geq| a |-| b |$
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Establish the following vector inequalities geometrically or otherwise:
$(a)$ $\quad| a + b | \leq| a |+| b |$
$(b)$ $\quad| a + b | \geq| a |-| b |$
$(c)$ $\quad| a - b | \leq| a |+| b |$
$(d)$ $\quad| a - b | \geq| a |-| b |$
When does the equality sign above apply?