Let ` vec a , vec b, vec c` be three non-zero vectors such that any two of them are non-collinear. If ` vec a+2 vec b` is collinear with ` vec ca n d vec b+3 vec c` is collinear with ` vec a` then prove that ` vec a+2 vec b+6 vec c= vec0`

Explanation

To prove that \( \vec{a} + 2\vec{b} + 6\vec{c} = \vec{0} \), we start with the given conditions regarding the vectors \( \vec{a}, \vec{b}, \) and \( \vec{c} \).

### Step 1: Establish the collinearity conditions
From the problem, we know:
1. \( \vec{a} + 2\vec{b} \) is collinear with \( \vec{c} \).
2. \( \vec{b} + 3\vec{c} \) is collinear with \( \vec{a} \).

Using the property of collinearity, we can express these conditions mathematically:
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