Question
If $\vec{\text{A}},\vec{\text{B}},\vec{\text{C}}$ are mutually perpendicular, show that $\vec{\text{C}}\times(\vec{\text{A}}\times\vec{\text{B}})=0.$ Is the converse true?
✓
Answer
Given that $\vec{\text{A}},\vec{\text{B}}$ and $\vec{\text{C}}$ are mutually perpendicular$\vec{\text{A}}\times\vec{\text{B}}$ is a vector which direction is perpendicular to the plane containing $\vec{\text{A}}$ and $\vec{\text{B}}.$ Also $\vec{\text{C}}$ is perpendicular to $\vec{\text{A}}$ and $\vec{\text{B}}$$\therefore$ Angle between $\vec{\text{C}}$ and $\vec{\text{A}}\times\vec{\text{B}}$ is $0^{\circ}$ or $180^{\circ}$
So, $\vec{\text{C}}\times(\vec{\text{A}}\times\vec{\text{B}})=0$ The converse is not true. For example, if two of the vector are parallel, then also$\vec{\text{C}}\times(\vec{\text{A}}\times\vec{\text{B}})=0$
So, they need not be mutually perpendicular.
So, $\vec{\text{C}}\times(\vec{\text{A}}\times\vec{\text{B}})=0$ The converse is not true. For example, if two of the vector are parallel, then also$\vec{\text{C}}\times(\vec{\text{A}}\times\vec{\text{B}})=0$
So, they need not be mutually perpendicular.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
The energy levels of an atom are as shown below:
- Which of them will result in the transition of a photon of wavelength 275 nm?
- Which transition corresponds to emission of radiation of maximum wavelength?
What is amplitude modulation ? Draw the simple block diagram of a modulator to get amplitude modulated wave.
- Define self $-$ inductance. Write its $SI$ units.
- A long solenoid with $15$ turns per $\ cm$ has a small loop of area $2.0 \ cm^2$ placed inside the solenoid normal to its axis. If the current carried by the solenoid changes steadily from $ 2.0 A$ to $4.0 A$ in $0.1 s,$ what is the induced emf in the loop while the current is changing?
Define the terms ‘stopping potential’ and ‘threshold frequency’ in relation to photoelectric effect. How does one determine these physical quantities using Einstein’s equation?
Find the average velocity of a projectile between the instants it crosses half the maximum height. It is projected with a speed u at an angle $\theta$ with the horizontal.
→What is Remote Sensing ? Mention two of its applications.
Two charges $5nC$ and $-2nC$ are placed at points $(5\ cm, 0, 0)$ and $(23\ cm, 0, 0)$ in a region of space where there is no other external field. Calculate the electrostatic potential energy of this charge system.
→The number of electrons in an insulator is of the same order as the number of electrons in a conductor. What is then the basic difference between a conductor and an insulator?
What are the advantages of the null$-$point method in a Wheatstone bridge? What additional measurements would be required to calculate $R_{unknown}$ by any other method?
→According to the station clocks, two babies are born at the same instant, one in Howrah and other in Delhi.
- Who is elder in the frame of 2301 Up Rajdhani Express going from Howrah to Delhi?
- Who is elder in the frame of 2302 Dn Rajdhani Express going from Delhi to Howrah.