Consider two non-zero vectors a and b in three-dimensional space.
a. (i) State the relation between a⋅b, ∣a∣, ∣b∣ and the angle θ between the vectors.
b. (ii) Prove that if a⋅b=∣a∣∣b∣, then a and b are parallel and point in the same direction.
c. (iii) Prove that if a⋅b=−∣a∣∣b∣, then a and b are parallel but point in opposite directions.
d. (iv) Give one concrete numerical example in R3 to illustrate each of the results obtained in parts (ii) and (iii).