Question
Establish the following vector inequalities geometrically or otherwise: $|\text{a}-\text{b}|\ge||\text{a}|-|\text{b}||$When does the equality sign above apply?
✓
Answer
Let two vectors $\vec{\text{a}}$ and $\vec{\text{b}}$ be represented by the adjacent sides of a parallelogram PORS, as shown in the given figure. 
The following relations can be written for the given parallelogram. (OS + PS) > OP .....(i) OS < (OP - PS) .....(ii) $|\vec{\text{a}}-\vec{\text{b}}|>|\vec{\text{a}}|-|\vec{\text{b}}|\ ...(\text{iii})$ The quantity on the LHS is always positive and that on the RHS can be positive or negative. To make both quantities positive, we take modulus on both sides as: $||\vec{\text{a}}-\vec{\text{b}}||>||\vec{\text{a}}|-|\vec{\text{b}}||$ $|\vec{\text{a}}-\vec{\text{b}}|>||\vec{\text{a}}|-|\vec{\text{b}}||\ ...(\text{iv})$ If the two vectors act in a straight line but in opposite directions, then we can write: $|\vec{\text{a}}-\vec{\text{b}}|=||\vec{\text{a}}|-|\vec{\text{b}}||\ ...(\text{v})$ Combining equations (iv) and (v), we get: $|\vec{\text{a}}-\vec{\text{b}}|\ge||\vec{\text{a}}|-|\vec{\text{b}}||$
The following relations can be written for the given parallelogram. (OS + PS) > OP .....(i) OS < (OP - PS) .....(ii) $|\vec{\text{a}}-\vec{\text{b}}|>|\vec{\text{a}}|-|\vec{\text{b}}|\ ...(\text{iii})$ The quantity on the LHS is always positive and that on the RHS can be positive or negative. To make both quantities positive, we take modulus on both sides as: $||\vec{\text{a}}-\vec{\text{b}}||>||\vec{\text{a}}|-|\vec{\text{b}}||$ $|\vec{\text{a}}-\vec{\text{b}}|>||\vec{\text{a}}|-|\vec{\text{b}}||\ ...(\text{iv})$ If the two vectors act in a straight line but in opposite directions, then we can write: $|\vec{\text{a}}-\vec{\text{b}}|=||\vec{\text{a}}|-|\vec{\text{b}}||\ ...(\text{v})$ Combining equations (iv) and (v), we get: $|\vec{\text{a}}-\vec{\text{b}}|\ge||\vec{\text{a}}|-|\vec{\text{b}}||$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
A uniform wheel of radius R is set into rotation about its axis at an angular speed $\omega.$ This rotating wheel is now placed on a rough horizontal surface with its axis horizontal. Because of friction at the contact, the wheel accelerates forward and its rotation decelerates till the wheel starts pure rolling on the surface. Find the linear speed of the wheel after it starts pure rolling.
→An insulated container containing monoatomic gas of molar mass m is moving with a velocity $v_0$. If the container is suddenly stopped, find the change in temperature.
→A stone of mass m tied to the end of a string revolves in a vertical circle of radius R. The net forces at the lowest and highest points of the circle directed vertically downwards are: [Choose the correct alternative]
$T_1$ and $v_1$ denote the tension and speed at the lowest point. $T_2$ and $v_2$ denote corresponding values at the highest point.
→|
|
Lowest Point
|
Highest Point
|
|
(a)
|
$mg - T_1$
|
$mg + T_2$
|
|
(b)
|
$mg + T_1$
|
$mg - T_2$
|
|
(c)
|
$mg + T_1 - (mv_1^2)/R$
|
$mg - T_2 + (mv_1^2)/R$
|
|
(d)
|
$mg - T_1 - (mv_1^2)/R$
|
$mg + T_2 + (mv_1^2)/R$
|
A $5.0\mu\text{F}$ capacitor is charged to $12V$. The positive plate of this capacitor is now connected to the negative terminal of a $12V$ battery and vice versa. Calculate the heat developed in the connecting wires.
→The acceleration due to gravity on the surface of moon is $1.7m s^{-2}$. What is the time period of a simple pendulum on the surface of moon if its time period on the surface of earth is $3.5$ s? (g on the surface of earth is $9.8m s^{-2}$)
→What is viscosity? What are the factors affecting viscous force in a liquid flowing in a tube? Derive the relation for the velocity upto which the liquid can have streamlined flow.
A turn of radius 20m is banked for the vehicles going at a speed of 36 km/h. If the coefficient of static friction between the road and the tyre is 0.4, what are the possible speeds of a vehicle so that it neither slips down nor skids up?
Can the work by kinetic friction on an object be positive? Zero?
An adiabatic cylindrical tube of cross-sectional area $1cm^2$ is closed at one end and fitted with a piston at the other end. The tube contains $0.03g$ of an ideal gas. At 1atm pressure and at the temperature of the surrounding, the length of the gas column is $40cm$. The piston is suddenly pulled out to double the length of the column. The pressure of the gas falls to $0.355atm$. Find the speed of sound in the gas at atmospheric temperature.
→Point masses $m_1$ and $m_2$ are placed at the opposite ends of a rigid rod of length L, and negligible mass. The rod is to be set in rotation about an axis perpendicular to it. Find the position on this rod through which the axis should pass in order that the work required to set the rod in rotation with angular velocity $\omega_0$ should be minimum.
→