Answer the following questions related to the scalar (dot) product of vectors.
(i) State the algebraic (component–wise) definition of the dot product of two vectors in three–dimensional space.
(ii) List any two properties of the dot product that follow directly from this definition.
(iii) Let a=⟨3,−2,1⟩ and b=⟨4,0,−5⟩. Verify, by direct calculation, that the dot product is commutative for this pair of vectors.