What does $v = v_0 + t_1(v_1 - v_0) + t_2(v_2 - v_0)$ parameterize where $t_i$s are scalars and $v_i$s are vectors?

Question

On the one hand $v$ looks like it describes a plane. On the other hand, $v_0 + t_1(v_1 — v_0)$ describes a line in $3$-space. Since we need two vectors(?) to describe a line, $t_2(v_2 - v_0)$ is redundant, meaning $v$ describes a line. Can you elaborate on why it's right/wrong?

Answer 1

It describes a plane unless $(v_1-v_0)$ and $(v_2-v_0)$ are linearly dependent. If they are dependent, then you can write $(v_2-v_0)=\lambda(v_1-v_0)$ and so the equation becomes $v_0+t_3(v_1-v_0)$ where $t_3=t_1+\lambda t_2$. That is, they only define points lying on the line in the direction of $v_1-v_0$ passing through the point $v_0$.

If they are linearly independent, then for any non zero value of $t_2$ the resulting point would lie off the original line.

Answer 2

In what follows, I assume none of the vectors is the zero vector.

(i) Assuming the vectors $\vec{v}_0, \vec{v}_1, $ and $\vec{v}_2$ are linearly independent. The coordinates of a valid point are given, parametrically, as $$ \left(\begin{array}{c}1-t_1-t_2\\t_1\\t_2\end{array}\right)\,. $$

In cartesian coordinates, this defines the plane $x+y+z{}={}1$.

(ii) If there is linear dependence between, say, $\vec{v}_0$ and $\vec{v}_1$, then the coordinates of a representative vector become (for $\vec{v}_1{}={}\lambda\vec{v}_0$) $$ \left(\begin{array}{c}1+t_1(\lambda-1)-t_2\\t_2\end{array}\right)\,, $$

where $\lambda\ne 0$. In cartesian coordinates, if $\lambda=1$, then this represents the line $x+y=1$. Otherwise, this represents the entire $xy$ plane. By symmetry, analogous considerations follow for linear dependence between $\vec{v}_0$ and $\vec{v}_2$.

If, instead, $\vec{v}_2{}={}\lambda\vec{v}_1$, then the points of interest now look like $$ \left(\begin{array}{c}1-t_1-t_2\\t_1+\lambda t_2\end{array}\right)\,, $$ where $\lambda\ne 0$. Like the previous case, this either represents a line (when $\lambda=1$) or, otherwise, the entire $xy$ plane.

(iii) Finally, if all the vectors are parallel, then this represents either the vector $\vec{v}_0$, or the line spanned by $\vec{v}_0$.