Multivariable calculus

Question

Find two non-zero vectors u and v in R³ such that {s u+ t v | s, t ∈ R} is a line in R³.

Answer 1

If two vectors point in the same direction, then any scale multiples of those two vectors will lie on the same line.

So we just need to choose two vectors pointing in the same direction. One option would be u = (1, 0, 0) and

v = (2, 0, 0). These two vectors point in the same direction (they are just scale multiples of each other), so they also lie on the same line. This would be a valid answer.

In contrast, if the two vectors we choose are linearly independent (not scale multiples of each other), then they would point in different directions and would span a plane in R³.