Equations defining span of vectors

Question

v1 = (1,-2,t), v2 = (t,t²,t²). For each fixed t, give equations defining the span of v1 and v2.

Could someone please help answer this question

Answer 1

Calculate the coordinates of $w:=v_1\times v_2$.
If that's not zero, then your equation for $x=(x_1,x_2,x_3)$ will be $$w_1x_1+w_2x_2+w_3x_3=0$$ since $x\in\mathrm{span}(v_1, v_2)\ \iff \ x\perp w\ \iff \ \langle w,x\rangle=0$.

Answer 2

Just calculate a basis for the kernel of the matrix which row vectors are $v_1$ and $v_2$. I get $(-2,-1,0)^t$ and $(0,0,1)^t$ for $t=0$, hence in this case the span is given by the equations $-2x-y=0$ and $z=0$; for $t=-2$ I get $x-2y=0$ and $2x+z=0$.

Now if $\notin\{0,-2\}$ I found the equation $tx+ty-z=0$.