Geometric interpretation of Linear independence

Question

In one of the standard text of Linear algebra, I Saw the below statement:

Three vectors in $\mathbb{R}^3$ are linearly independent if and only if they do not lie in the same plane when they have their initial points at the origin. Otherwise at least one would be the linear combination of other two.

Is following example contradicts this statement?

Consider the vectors $v_1= (1,0,0), v_2=(0,2,1), v_3=(2,0,0)$ then we can easily show that, these vectors are linearly dependent (since $v_1$ and $v_2$ are scalar multiple of each other) also we can see that, vectors $v_1, v_2, v_3$ do not lie in same plane.

They all lie in the same plane spanned by $v_1$ and $v_2$. Note that $v_3 = 2v_1 + 0v_2$. — CyclotomicField, Jun 26 '22 at 04:30
@CyclotomicField sir, by writing $v_3= 2v_1+ 0v_2$, how can we say they lie in same plane? I can see $v_1, v_3$ are lie in same plane (since both lie on x-axis) but but how $v_2$ belongs to same plane? — Akash Patalwanshi, Jun 26 '22 at 04:40
Consider an even simpler example, $v_1=(1,0,0),v_2=(0,1,0), v_3=(2,0,0)$. This is obviously linearly dependent, for the same reason as you mentioned. Now, clearly these three vectors lie in the same plane (namely the $x$-$y$ plane). In any case, $v_3$ lies on the same line as $v_1$, so it definitely lies inside any plane which contains $v_1$. — peek-a-boo, Jun 26 '22 at 05:00
@AkashPatalwanshi a plane is the span of two linearly independent vectors. Since $v_1$ and $v_2$ are linearly independent, they span a plane. Since $v_3$ is a linear combination of those two vectors, it lays in that plane. — CyclotomicField, Jun 26 '22 at 05:04
Notice all three vectors lie in the plane $0x-y+2z=0$. [Reason: The origin $(0,0,0)$ and the heads of all three vectors, namely $(1,0,0)$, $(0,2,1)$, and $(2,0,0)$ all satisfy this equation and so all lie in the plane described by this equation.] — Selrach Dunbar, Jun 26 '22 at 05:15
For a set of three vectors, why is linear dependence is equivalent to coplanarity? — ryang, Jun 26 '22 at 09:04

Geometric interpretation of Linear independence

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