Show that the vectors ` vec a-2 vec b+3 vec c , vec a-3 vec b+5 vec ca n d-2 vec a+3 vec b-4 vec c` are coplanar, where ` vec a , vec b , vec c` are non-coplanar.

Explanation

To show that the vectors \( \vec{a} - 2\vec{b} + 3\vec{c} \), \( \vec{a} - 3\vec{b} + 5\vec{c} \), and \( -2\vec{a} + 3\vec{b} - 4\vec{c} \) are coplanar, we can use the condition that three vectors \( \vec{p}, \vec{q}, \vec{r} \) are coplanar if one of the vectors can be expressed as a linear combination of the other two.

Let's denote:
- \( \vec{p} = \vec{a} - 2\vec{b} + 3\vec{c} \)
- \( \vec{q} = \vec{a} - 3\vec{b} + 5\vec{c} \)
- \( \vec{r} = -2\vec{a} + 3\vec{b} - 4\vec{c} \)

We need to check if there exist scalars \( \alpha \) and \( \beta \) such that:
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