Explain the parallelogram method for vector addition. Also,explain how this is comparable to the triangle method.
(N/A) $1$. Consider two vectors $\vec{A}$ and $\vec{B}$ as shown in figure $(a)$.
$2$. To add them using the parallelogram method,place the tails of both vectors at a common point $O$ as shown in figure $(b)$.
$3$. Construct a parallelogram $OPSQ$ such that $\vec{A}$ and $\vec{B}$ are adjacent sides. The diagonal $OS$ starting from $O$ represents the resultant vector $\vec{R} = \vec{A} + \vec{B}$.
$4$. In the triangle method,we place the tail of $\vec{B}$ at the head of $\vec{A}$. The resultant is the vector from the tail of $\vec{A}$ to the head of $\vec{B}$,as shown in figure $(c)$.
$5$. Since the side $PS$ in the parallelogram is parallel and equal to $OQ$ (which is $\vec{B}$),the triangle $OPS$ in the parallelogram method is identical to the triangle formed in the triangle method.
$6$. Thus,both methods yield the same resultant vector $\vec{R}$.
$7$. The magnitude of the resultant vector satisfies the triangle inequality: $|\vec{R}| \leq |\vec{A}| + |\vec{B}|$.
$2$. To add them using the parallelogram method,place the tails of both vectors at a common point $O$ as shown in figure $(b)$.
$3$. Construct a parallelogram $OPSQ$ such that $\vec{A}$ and $\vec{B}$ are adjacent sides. The diagonal $OS$ starting from $O$ represents the resultant vector $\vec{R} = \vec{A} + \vec{B}$.
$4$. In the triangle method,we place the tail of $\vec{B}$ at the head of $\vec{A}$. The resultant is the vector from the tail of $\vec{A}$ to the head of $\vec{B}$,as shown in figure $(c)$.
$5$. Since the side $PS$ in the parallelogram is parallel and equal to $OQ$ (which is $\vec{B}$),the triangle $OPS$ in the parallelogram method is identical to the triangle formed in the triangle method.
$6$. Thus,both methods yield the same resultant vector $\vec{R}$.
$7$. The magnitude of the resultant vector satisfies the triangle inequality: $|\vec{R}| \leq |\vec{A}| + |\vec{B}|$.
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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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