Is a physical quantity, having a direction, necessarily a vector?

No, all physical quantities having  direction is not a vector. 
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Give three examples of vector quantities.


Force, impulse and momentum.
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What are the basic characteristics that a quantity must possess so that it may be a vector quantity?

A quantity must possess the direction and must follow the vector axioms. Any quantity that follows the vector axioms are classified as vectors. 


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Which of the followings is/are vector quantities: work, power, acceleration, mass, momentum?

Acceleration and momentum are vector quantities. 
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Give three examples of scalar quantities.

Mass, temperature and energy
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