I'm trying to understand when we can express vectors as planes vs lines when they are written in parametric form. I understand that, normally, if you have two or three vectors that are not multiples of each other they will span a plane. However, I would like to translate this to an example I made up in parametric form to ensure I understand this correctly:
Say the parametric form is: [8 -4 0] + x2[-1 1 -1]. Geometrically, we can describe this as a a line in R3 through [8 -4 0] and parallel to vector [-1 1 -1]. It spans a line simply because the first vector is simply a point and x2 spans a line because x2 can vary?
If we look at this same example, but add x3[2, 7, 9] (something that isn't a multiple of x2) to our original parametric equation, then would this span a plane in R3? My logic is because now we have 2 vectors that span 2 lines that are not multiples of each other, effectively spanning a plane?
Does this logic make sense? Thank you!
Best Answer
Yes, that sounds right. In general, if we have $k$ linearly independent vectors $v_1, v_2, …, v_k$ and some vector $v$, then the set $\{v + a_1v_1 + a_2v_2 … + a_kv_k: a_1, a_2, …, a_k \in \mathbb{R} \}$ is just the set of all vectors in the span of $v_1, v_2, …, v_k$, but shifted by the vector $v$. For example, if we have two linearly independent vectors $v_1, v_2$ in $\mathbb{R}^3$, these will span a plane, and adding on $v$ just shifts this plane to the point $v$. In your example, $v = (8, -4, 0), v_1 = (-1, 1, -1), v_2 = (2,7,9)$.