Dot Product of Difference

By the fundamental properties of reciprocal systems of vectors:

$$\vec{a} \cdot \vec{a'} = 1, \quad \vec{b} \cdot \vec{b'} = 1, \quad \vec{c} \cdot \vec{c'} = 1$$

$$\vec{a} \cdot \vec{b'} = \vec{a} \cdot \vec{c'} = \vec{b} \cdot \vec{a'} = \vec{b} \cdot \vec{c'} = \vec{c} \cdot \vec{a'} = \vec{c} \cdot \vec{b'} = 0$$

Let's expand the given expression term by term using linearity of the dot product:

1. First term:

$$(\vec{a} + 2\vec{b} - \vec{c}) \cdot \vec{a'} = \vec{a}\cdot\vec{a'} + 2(\vec{b}\cdot\vec{a'}) - \vec{c}\cdot\vec{a'} = 1 + 0 - 0 = 1$$

2. Second term:

$$(\vec{b} + 2\vec{c} - \vec{a}) \cdot \vec{b'} = \vec{b}\cdot\vec{b'} + 2(\vec{c}\cdot\vec{b'}) - \vec{a}\cdot\vec{b'} = 1 + 0 - 0 = 1$$

3. Third term:

$$(\vec{c} + 2\vec{a} - \vec{b}) \cdot \vec{c'} = \vec{c}\cdot\vec{c'} + 2(\vec{a}\cdot\vec{c'}) - \vec{b}\cdot\vec{c'} = 1 + 0 - 0 = 1$$

Adding them all together:

$$\text{Value} = 1 + 1 + 1 = 3$$