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Pick out the only vector quantity in the following list:

Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge.

Impulse = change in momentum = force × time.

Momentum and force are both vector quantities. Hence, impulse is the vector quantity.
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Pick out the two scalar quantities in the following list:

force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity.

Work and current are scalar quantities.


The dot product of force and displacement gives us the amount of work done. And, we know that the dot product of two quantities is always a scalar. Therefore, work is a scalar physical quantity.

Current is described only by its magnitude. Its direction is not taken. Hence, it is a scalar quantity.
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Establish the vector inequalities geometrically or otherwise : 


| a + b | ≥ || a | −| b ||


Let two vectors a and b be represented by the adjacent sides of a parallelogram OMNP, as shown in the given figure. 

 
Here, we have: 

| OM | = | a | 

| MN | = | OP | = | b | 

| ON | = | a + b | 

In a triangle, each side is smaller than the sum of the other two sides. 

Therefore, in ΔOMN, we have,

ON + MN > OM 

ON + OM > MN 

| ON | > | OM - OP |   (∵ OP = MN) 

| a + b | > | | a | - | b | |                 ...(iv) 

If the two vectors a and b act along a straight line in the same direction, then we can write: 

| a + b | = | | a | - | b | |                 ...(v) 

Combining equations (iv) and (v), we get: 

| a + b | ≥ | | a | - | b | |  

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Establish the vector inequality:
(a) |b| ≤ |a| + |b|


Let two vectors a and b be represented by the adjacent sides of a parallelogram OMNP, as shown in the given figure.





Here, we can write: 

OM = | a |                                 ...(i) 
    
MN = OP = | b |                         ...(ii) 

ON = | a + b |                            ...(iii) 

In a triangle, each side is smaller than the sum of the other two sides. 

Therefore, in ΔOMN, we have,

ON < (OM + MN) 

| a + b | < | a | + | b |                 ...(iv) 

If the two vectors a and b act along a straight line in the same direction, then we can write: 

| a + b | = | a | + | b |                  ... (v) 

Combining equations (iv) and (v), we get: 

| a + b | ≤ | a | + | b |
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Three girls skating on a circular ice ground of radius 200 m start from point P on the edge of the ground and reach a point Q diametrically opposite to P following different path as shown. What is the magnitude of displacement vector for each? For which girl is this equal to the actual length of path skate? 
                      

Given, 

Radius of the skating ground = 200 m 

The initial and final position of all grls are same. Hence, displacement of each girl is same. 

∴    Displacement  

Magnitude of displacemnt is given by,

 

From the figure, girl B skates along the straight path PQ. Therefore, for girl B, the magnitude of displacement and distance will be equal. 
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