Prove the associative law of vector addition.
(N/A) To prove the associative law of vector addition,$(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})$,consider three vectors $\vec{A}, \vec{B}$,and $\vec{C}$ represented by the sides of a polygon.
Let $\vec{A} = \overrightarrow{OP}$,$\vec{B} = \overrightarrow{PQ}$,and $\vec{C} = \overrightarrow{QR}$.
Using the triangle law of vector addition in $\Delta OPQ$:
$\vec{A} + \vec{B} = \overrightarrow{OP} + \overrightarrow{PQ} = \overrightarrow{OQ}$.
Now,adding $\vec{C} = \overrightarrow{QR}$ to both sides:
$(\vec{A} + \vec{B}) + \vec{C} = \overrightarrow{OQ} + \overrightarrow{QR} = \overrightarrow{OR} \quad \dots (i)$.
Next,consider $\Delta PQR$:
$\vec{B} + \vec{C} = \overrightarrow{PQ} + \overrightarrow{QR} = \overrightarrow{PR}$.
Now,adding $\vec{A} = \overrightarrow{OP}$ to both sides:
$\vec{A} + (\vec{B} + \vec{C}) = \overrightarrow{OP} + \overrightarrow{PR} = \overrightarrow{OR} \quad \dots (ii)$.
From equations $(i)$ and $(ii)$,we get:
$(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})$.
This proves the associative law of vector addition.
Let $\vec{A} = \overrightarrow{OP}$,$\vec{B} = \overrightarrow{PQ}$,and $\vec{C} = \overrightarrow{QR}$.
Using the triangle law of vector addition in $\Delta OPQ$:
$\vec{A} + \vec{B} = \overrightarrow{OP} + \overrightarrow{PQ} = \overrightarrow{OQ}$.
Now,adding $\vec{C} = \overrightarrow{QR}$ to both sides:
$(\vec{A} + \vec{B}) + \vec{C} = \overrightarrow{OQ} + \overrightarrow{QR} = \overrightarrow{OR} \quad \dots (i)$.
Next,consider $\Delta PQR$:
$\vec{B} + \vec{C} = \overrightarrow{PQ} + \overrightarrow{QR} = \overrightarrow{PR}$.
Now,adding $\vec{A} = \overrightarrow{OP}$ to both sides:
$\vec{A} + (\vec{B} + \vec{C}) = \overrightarrow{OP} + \overrightarrow{PR} = \overrightarrow{OR} \quad \dots (ii)$.
From equations $(i)$ and $(ii)$,we get:
$(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})$.
This proves the associative law of vector addition.
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Similar Questions
Given below in Column $-I$ are the relations between vectors $\vec a$,$\vec b$,and $\vec c$,and in Column $-II$ are the orientations of $\vec a$,$\vec b$,and $\vec c$ in the $XY-$ plane. Match the relation in Column $-I$ to the correct orientations in Column $-II$.
Column $-I$ Column $-II$ $(a) \vec a + \vec b = \vec c$ $(i)$ Vector $\vec a$ is along $+Y$,$\vec c$ is along $+X$,and $\vec b$ connects the origin to the tip of $\vec c$ $(b) \vec a - \vec c = \vec b$ $(ii)$ Vector $\vec a$ is along $+X$,$\vec b$ is along $+Y$,and $\vec c$ connects the origin to the tip of $\vec b$ $(c) \vec b - \vec a = \vec c$ $(iii)$ Vector $\vec c$ is along $+X$,$\vec a$ is along $+Y$,and $\vec b$ connects the tip of $\vec c$ to the tip of $\vec a$ $(d) \vec a + \vec b + \vec c = 0$ $(iv)$ Vector $\vec a$ is along $-X$,$\vec b$ is along $-Y$,and $\vec c$ connects the origin to the tip of $\vec b$
| Column $-I$ | Column $-II$ |
| $(a) \vec a + \vec b = \vec c$ | $(i)$ Vector $\vec a$ is along $+Y$,$\vec c$ is along $+X$,and $\vec b$ connects the origin to the tip of $\vec c$ |
| $(b) \vec a - \vec c = \vec b$ | $(ii)$ Vector $\vec a$ is along $+X$,$\vec b$ is along $+Y$,and $\vec c$ connects the origin to the tip of $\vec b$ |
| $(c) \vec b - \vec a = \vec c$ | $(iii)$ Vector $\vec c$ is along $+X$,$\vec a$ is along $+Y$,and $\vec b$ connects the tip of $\vec c$ to the tip of $\vec a$ |
| $(d) \vec a + \vec b + \vec c = 0$ | $(iv)$ Vector $\vec a$ is along $-X$,$\vec b$ is along $-Y$,and $\vec c$ connects the origin to the tip of $\vec b$ |
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