Three vectors are on the same plane

[mathmari]

Hey!

We want to show that if $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are on the same plane, then there are $A, B, C$ not all $0$ such that $A \vec a+B \vec b+C \vec c=\vec 0$.
The solution is the following:

If $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are on the same plane, then

1. all are equal to $\overrightarrow{0}$ and then $1 \cdot \overrightarrow{a} + 1 \cdot \overrightarrow{b}+ 1 \cdot \overrightarrow{c}=\overrightarrow{0}$
   
   
2. all are on the same line and one is not the $\overrightarrow{0}$
   
   for example, $\overrightarrow{a} \neq \overrightarrow{0}$, so $\overrightarrow{b}=m \overrightarrow{a}, \overrightarrow{c}=n\overrightarrow{a}$ and then $-(m+n)\overrightarrow{a}+1\cdot \overrightarrow{b}+1 \overrightarrow{c}=\overrightarrow{0}$
3. two are not the $\overrightarrow{0}$ and don't belong to the same line
   
   for example, the $\overrightarrow{b}, \overrightarrow{c}$, then $\overrightarrow{a}=x \overrightarrow{b}+y \overrightarrow{c}$ and then $1 \cdot \overrightarrow{a}+(-x) \overrightarrow{b}+(-y) \overrightarrow{c}=\overrightarrow{0}$.

Could you explain to me why we take these cases?

 

[Evgeny.Makarov]

Could you explain to me why we take these cases?

Because it leads to a valid proof.

 

[mathmari]

At the beginning we take the case if all the three verctors are the zero vector. Then when at least one of the three vectors is not the zero vector we take the cases if the vectors are at the same line or not. Is this correct?? (Wondering)

Why do we take at the second case: "all are on the same line and one is not the $\overrightarrow{0}$" that only one is not the zero vector and at the third case "two are not the $\overrightarrow{0}$ and don't belong to the same line" that two vectors are not the zero vector?? (Wondering) Can it not be that one is not the $\overrightarrow{0}$ and they don't belong to the same line?? (Wondering)