Final answer:
The imaginary line extending infinitely along a vector through the tip and tail is called a vector line, which is integral to visualizing a vector's magnitude and direction.
Step-by-step explanation:
The imaginary line that extends infinitely along a vector through both the tip and tail is termed a vector line. When considering a vector, it is represented as an arrow that has a magnitude, which is its length, and a direction, indicated by the arrowhead. The starting point or tail is where the vector begins, and the head or tip is where it ends. As you correctly stated, at either end of the line or even considering the middle of the line (which can also be referred to as 'the origin' in broader mathematical terms), that line representing the vector continues infinitely in both directions. This is similar to how we visualize the field of an infinite straight wire in physics. Whether the vectors are drawn head-to-head, tail-to-tail, or in the head-to-tail method for vector addition, the extended line through the vectors is an important concept for understanding how vectors work in space.