Motion in a Plane — Physics STD 11 Science — Question

To show that vector addition is associative, we consider addition of three vectors $\vec{\text{A}},\vec{\text{B}}$ and $\vec{\text{C}}$ in two different manners. Let us first add $\vec{\text{A}}$ and $\vec{\text{B}}$ to obtain a vector $\overrightarrow{\text{KM}}$ and then add $\vec{\text{C}}$ to it so as to get the resultant vector $\overrightarrow{\text{KN}}.$ It means that

$(\vec{\text{A}}+\vec{\text{B}})+\vec{\text{C}}=\overrightarrow{\text{KN}}=\vec{\text{R}}\dots(\text{i})$

Again we add $\vec{\text{B}}$ and $\vec{\text{C}}$ to obtain a vector $\overrightarrow{\text{LN}}.$ Now to vector $\vec{\text{A}}$ add $\overrightarrow{\text{LN}}$ so as to get a resultant $\overrightarrow{\text{KN}}=\vec{\text{R}}$ as shown in Fig. It means that,

$(\vec{\text{A}})+(\vec{\text{B}}+\vec{\text{C}})=\overrightarrow{\text{KL}}+\overrightarrow{\text{LN}}=\overrightarrow{\text{KN}}=\vec{\text{R}}\dots(\text{ii})$

From (i) and (ii), it is clear that,

$(\vec{\text{A}}+\vec{\text{B}})+\vec{\text{C}}=\vec{\text{A}}+(\vec{\text{B}}+\vec{\text{C}})$

That is the vector addition is associative.