For three non-zero vectors `vec(a),\vec(b) " and"vec(c )`, prove that `[(vec(a)-vec(b))\ \ (vec(b)-vec(c))\ \ (vec(c )-vec(a))]=0`

Explanation

To prove that \([( \vec{a} - \vec{b})\ \ (\vec{b} - \vec{c})\ \ (\vec{c} - \vec{a})] = 0\) for three non-zero vectors \(\vec{a}, \vec{b}, \vec{c}\), we will use the properties of the scalar triple product.

### Step-by-step Solution:

1. **Define the Vectors**: Let \(\vec{u} = \vec{a} - \vec{b}\), \(\vec{v} = \vec{b} - \vec{c}\), and \(\vec{w} = \vec{c} - \vec{a}\). We need to show that \([\vec{u}\ \ \vec{v}\ \ \vec{w}] = 0\).

2. **Express the Scalar Triple Product**: The scalar triple product \([\vec{u}\ \ \vec{v}\ \ \vec{w}]\) can be expressed as:
\[
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