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If a vector A is multiplied by a real number (say n), the vector of same nature is obtained but its magnitude is n times that of A.

According to the Heisenberg Uncertainty Principle, it is impossible to determine the exact spin of an electron at any given moment. The Uncertainty Principle states that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa. Spin is a form of intrinsic angular momentum, which means it does not correspond to any specific position in space, and therefore cannot be measured precisely at the same time as its position.

However, while we...

At dawn and dusk, large amount of blue and violet light(shorter wavelength) has been scattered so, the light that is recieved by an observer is mostly of a longer wavelength and therefore appears to be red.

Log_2aa=x    then, a=(2a)^x  ......(1)

Log_3a2a=y    then,2a=(3a)^y ......(2)

Log_4a 3a=z  then, 3a=(4a)^z ......(3)

So, 

a=(2a)^x  [from (1)]

Or, a=(3a)^xy    [from(2)]

Or, a=(4a)^xyz     [from(3)]

Multiplying both sides by 4a,

4a.a=4a.(4a)^xyz  

Or,(2a)² =(4a)^{xyz + 1} 

Or,(3a)^2y =(4a)^xyz+1 

Or,(4a)^2yz =(4a)^xyz+1 

Or, 2yz = xyz+1 .proved.