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Model Paper 3 — Physics STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 SciencePhysicsModel Paper 35 Marks
Question
What is meant by resolution of a vector? Prove that a vector can be resolved along two given directions in one and only one way.
✓

Answer

Resolution of a vector. It is the process of splitting a vector into two or more vectors in such a way that their combined effect is same as that of the given vector. The vectors into which the given vector is splitted are called component vectors. A component of a vector in any direction gives a measure of the effect of the given vector in that direction. The resolution of a vector is just opposite to the process of vector addition.
Resolution of a vector along two given directions.
Suppose we wish to resolve a vector $\vec{R}$ in the direction of two coplanar and non-parallel vectors $\vec{A}$ and $\vec{B}$, as shown in Figure.
Image
Suppose $\overrightarrow{O Q}$ represent vector $\vec{R}$ in the directions of $\vec{A}$ and $\vec{B}$.
Q draw lines parallel to vectors $\vec{A}$ and $\vec{B}$ respectively to meet at point P. From triangle law of vector addition.
$
\overrightarrow{O Q}=\overrightarrow{O P}+\overrightarrow{P Q}
$
As $\overrightarrow{O P} \| \vec{A}$ therefore, $\overrightarrow{O P}=\lambda \vec{A}$
As $\overrightarrow{P Q} \| \vec{B}$ therefore, $\overrightarrow{P Q}=\mu \vec{B}$
Here $\lambda$ and $\mu$ are scalar. Hence
$
\vec{R}=\lambda \vec{A}+\mu \vec{B} \ldots \text { (i) }
$
Thus the vector $\vec{R}$ has been resolved in the direction of $\vec{A}$ and $\vec{B}$. Here $\lambda \vec{A}$ is the component of $\vec{R}$ in the direction $\vec{A}$ and $\mu \vec{B}$ is the component in the direction of $\vec{B}$.
Uniqueness of resolution. Let us assume that $\vec{R}$ can be resolved in the directions of $\vec{A}$ and $\vec{B}$ in another way.
Then $\vec{R}=\lambda^{\prime} \vec{A}+\mu^{\prime} \vec{B} \ldots$... (ii)
From equation (i) and (ii), we have
$
\begin{aligned}
& \lambda \vec{A}+\mu \vec{B}=\lambda^{\prime} \vec{A}+\mu^{\prime} \vec{B} \\
& \text { or }\left(\lambda-\lambda^{\prime}\right) \vec{A}=\left(\mu^{\prime}-\mu\right) \vec{B}
\end{aligned}
$
But $\vec{A}$ and $\vec{B}$ are non-zero vectors acting along different directions. The above equation is possible only if
$
\begin{aligned}
& \lambda-\lambda^{\prime}=0 \text { and } \mu^{\prime}-\mu=0 \\
& \text { or } \lambda^{\prime}=\lambda \text { and } \mu^{\prime}=\mu
\end{aligned}
$
Hence there is one and only one way in which a vector $\vec{R}$ can be resolved in the directions of vectors $\vec{A}$ and $\vec{B}$.

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