A vector \(\vec{A}\) is rotated by a small angle \(\Delta\theta\) radians \(\left(\Delta\theta<<1\right)\) to get a new vector \(\vec{B}.\) In that case \(\left|\vec{B}-\vec{A}\right|\) is :
- $0$
- $\left|\vec{A}\right|\left(1-\frac{\Delta\theta^{2}}{2}\right)$
- $\left|\vec{A}\right|\Delta\theta$
- $\left|\vec{B}\right|\Delta\theta-\left|\vec{A}\right|$
The Correct Option is C
Solution and Explanation
\(|\vec{B}-\vec{A}|=\sqrt{A^{2}+B^{2}+2 A B \cos (\pi-\Delta \theta)}\)
\(\Rightarrow\) since \(|\vec{A}|=|\vec{B}|=A,\)
\(\cos (\pi-\Delta \theta)=-\cos \Delta \theta\)
\(=\sqrt{2 A^{2}(1+\cos (\pi-\Delta \theta))}\)
\(=\sqrt{2 A^{2} 2 \cos ^{2}\left(\frac{\pi-\Delta \theta}{2}\right)}=2 A \cos \left(\frac{\pi}{2}-\frac{\Delta \theta}{2}\right)=2 A \sin \left(\frac{\Delta \theta}{2}\right)\)
Top Questions on Motion in a plane
- If a body projected with a velocity of 19.6 ms⁻¹ reaches a maximum height of 9.8 m, then the range of the projectile is (Neglect air resistance)
- TS EAMCET - 2025
- Physics
- Motion in a plane
- A cannon on a cliff 55 m above the ground fires a shell with a velocity of \(50\hat{i}+50\hat{j}\) ms\textsuperscript{−1}. The displacement vector of the shell when it hits the ground is (Acceleration due to gravity = 10 ms\textsuperscript{−2})
- TS EAMCET - 2025
- Physics
- Motion in a plane
- The minimum velocity of a projectile is 50% of its maximum velocity. If the minimum velocity of the projectile is $10~\text{m s}^{-1}$ then the time of flight of the projectile is (Acceleration due to gravity = $10~\text{m s}^{-2}$
- TS EAMCET - 2025
- Physics
- Motion in a plane
- A ball is thrown vertically upwards with an initial velocity of $ 20 \, \text{m/s} $. How high will the ball rise? (Take $ g = 10 \, \text{m/s}^2 $)
- MHT CET - 2025
- Physics
- Motion in a plane
- A person travelling on a straight line moves with a uniform velocity \( v_1 \) for a distance \( x \) and with a uniform velocity \( v_2 \) for the next \( \frac{3x}{2} \) distance. The average velocity in this motion is \( \frac{50}{7} \, \text{m/s} \). If \( v_1 \) is 5 m/s, then \( v_2 \) is ___ m/s.
- JEE Main - 2025
- Physics
- Motion in a plane
Questions Asked in JEE Main exam
- Let $\vec{a}=2\hat{i}-\hat{j}-\hat{k}$, $\vec{b}=\hat{i}+3\hat{j}-\hat{k}$ and $\vec{c}=2\hat{i}+\hat{j}+3\hat{k}$. Let $\vec{v}$ be the vector in the plane of $\vec{a}$ and $\vec{b}$, such that the length of its projection on the vector $\vec{c}$ is $\dfrac{1}{\sqrt{14}}$. Then $|\vec{v}|$ is equal to
- JEE Main - 2026
- Vector Algebra
A substance 'X' (1.5 g) dissolved in 150 g of a solvent 'Y' (molar mass = 300 g mol$^{-1}$) led to an elevation of the boiling point by 0.5 K. The relative lowering in the vapour pressure of the solvent 'Y' is $____________ \(\times 10^{-2}\). (nearest integer)
[Given : $K_{b}$ of the solvent = 5.0 K kg mol$^{-1}$]
Assume the solution to be dilute and no association or dissociation of X takes place in solution.- JEE Main - 2026
- Solutions
- The wavelength of photon 'A' is 400 nm. The frequency of photon 'B' is \(10^{16}\,\text{s}^{-1}\). The wave number of photon 'C' is \(10^{5}\,\text{cm}^{-1}\). The correct order of energy of these photons is:
- JEE Main - 2026
- General Chemistry
- Given below are two statements: Statement I: The increasing order of boiling point of hydrogen halides is HCl<HBr<HI<HF. Statement II: The increasing order of melting point of hydrogen halides is HCl<HBr<HF<HI. In the light of the above statements, choose the correct answer from the options given below:
- JEE Main - 2026
- General Chemistry
Inductance of a coil with \(10^4\) turns is \(10\,\text{mH}\) and it is connected to a DC source of \(10\,\text{V}\) with internal resistance \(10\,\Omega\). The energy density in the inductor when the current reaches \( \left(\frac{1}{e}\right) \) of its maximum value is \[ \alpha \pi \times \frac{1}{e^2}\ \text{J m}^{-3}. \] The value of \( \alpha \) is _________.
\[ (\mu_0 = 4\pi \times 10^{-7}\ \text{TmA}^{-1}) \]- JEE Main - 2026
- Ray optics and optical instruments
Concepts Used:
Motion in a Plane
It is a vector quantity. A vector quantity is a quantity having both magnitude and direction. Speed is a scalar quantity and it is a quantity having a magnitude only. Motion in a plane is also known as motion in two dimensions.
Equations of Plane Motion
The equations of motion in a straight line are:
v=u+at
s=ut+½ at2
v2-u2=2as
Where,
- v = final velocity of the particle
- u = initial velocity of the particle
- s = displacement of the particle
- a = acceleration of the particle
- t = the time interval in which the particle is in consideration