First, the word is "basis", not "base". If you are taking a course dealing with abstract vector spaces then you probably have, some time in the past, possibly in a physics course, drawn vectors as "directed line segments". That is, as a line segment (not "line"- a line segment has specific beginning and ending points while a line extends forever in both directions) with an "arrow head" drawn at one end.
Any two vectors in \(\displaystyle R^n\) and be basis vectors as long as one is not a multiple of the other. For example \(\displaystyle v_1= <1, 3>\) and \(\displaystyle v_2= <2, 4>\) are independent and so form a basis for \(\displaystyle R^2\). Take a sheet of paper and draw a Cartesian coordinate system on it. That is, draw two perpendicular lines crossing in the middle of the paper and label the origin, the point where the two lines intersect, (0, 0). Mark the two lines with coordinates, positive on one side of the origin, negative on the other. Most people make the axes horizontal and vertical, with positive on the right half of the horizontal line and on the top half of the vertical line, but that is not necessary. In any case, you now have a Cartesian coordinate system and so can label any point on the paper as "(x, y)". Mark the point (1, 3) and draw the line segment from (0, 0) to (1, 3) putting a little "arrow head" at (1, 3). That represents the vector <1, 3>. (Notice that I am using "<x, y>" for vectors and "(x, y)" for points in order to distinguish them. Do the same thing, marking the point (2, 4) in the coordinate system and draw the line segment from (0, 0) to (2, 4), putting a little "arrow head" at (2, 4).
Of course the whole point of a "basis" for a vector space is that any vector vector in the vector space can be written as a linear combination of the basis vectors. To show that here, geometrically, mark any point in the coordinate system, draw the line segment from (0, 0) to that point, draw a little "arrow head" at the point and label it "v". Extend the line segment representing \(\displaystyle v_1\) to a line extending infinitely in both directions. Draw a line parallel to \(\displaystyle v_2\) passing through the new point that formed v. Mark the point where they intersect (and they will precisely because the two basis vector are "independent") and draw a little "arrow head" on the line through \(\displaystyle v_1\) at that polnt and an arrow head on the line parallel to \(\displaystyle v_2\) at the point v. The first vector formed is a real number, \(\displaystyle a_1\) times \(\displaystyle v_1\), the second vector formed is a real number, \(\displaystyle a_2\) times \(\displaystyle v_2\) and the previous construction is their sum: \(\displaystyle v= a_1v_1+ a_2v_2\).
The same idea applies any basis for \(\displaystyle R^3\) with the added complication that a drawing on paper would be a two dimensional projection of a three dimensional situation on that paper. But you can at least visualize setting up a three dimensional xyz-coordinate system. Each given vector in \(\displaystyle R^3\) will be given as three numbers, <a, b, c>, which correspond to the point (a, b, c) in the coordinate system. Draw the line segment from (0, 0, 0) to (a, b, c) and draw a little "arrow head" at (a, b, c) to represent the vector <a, b, c>.
That's a very basic answer to your question, but then it is a very basic question!
