Prove that three vectors `vec(a),vec(b)andvec(c)` are coplanar if and only if there exists non-zero linear combination `xvec(a)+yvec(b)+zvec(c)=vec(0)`.

Explanation

To prove that three vectors \(\vec{a}, \vec{b}, \text{ and } \vec{c}\) are coplanar if and only if there exists a non-zero linear combination \(x\vec{a} + y\vec{b} + z\vec{c} = \vec{0}\), we will break down the proof into two parts:

1. **If \(\vec{a}, \vec{b}, \text{ and } \vec{c}\) are coplanar, then there exists a non-zero linear combination \(x\vec{a} + y\vec{b} + z\vec{c} = \vec{0}\).**
2. **If there exists a non-zero linear combination \(x\vec{a} + y\vec{b} + z\vec{c} = \vec{0}\), then \(\vec{a}, \vec{b}, \text{ and } \vec{c}\) are coplanar.**

### Step 1: Proving that coplanarity implies a non-zero linear combination

**Assumption:** Let \(\vec{a}, \vec{b}, \text{ and } \vec{c}\) be coplanar.
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