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Given $ =\mathbb{R}^3$ over the field $\mathbb{R}$ and we define the following operations: $$(, , ) + (, , ) = (, + , )$$

$$(, , ) = (, , )$$

Does $V$ over $\mathbb{R}$ with the defined operations is a vector space?

What do I need to check if I want to deduct whether this is a vector pace or not? Thanks

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    See http://www.math.niu.edu/~beachy/courses/240/06spring/vectorspace.html for a list of definitions and properties that will need to be checked – 2017-01-02

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What do I need to check if I want to deduct whether this is a vector pace or not

You need to check the definition of vector spaces.

In your particular case, you don't need to check all the axioms. Note that the new addition defined on $V$ is not commutative and thus $V$ with the new addition and the scalar multiplication is not a vector space.