已知 $\mathbf{a}$、$\mathbf{b}$ 和 $\mathbf{c}$ 是非零向量，且满足 $(\mathbf{a}+\mathbf{b})\times(\mathbf{a}+\mathbf{c}) = \mathbf{b} \times \mathbf{c}$，并且 $\mathbf{b} \ne \mathbf{c}$。求 $\mathbf{a}$、$\mathbf{b}$ 和 $\mathbf{c}$ 之间的关系。 另外已知 $\mathbf{a}$、$\mathbf{b}$ 和 $\mathbf{c}$ 满足方程 $\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}$，其中 $|\mathbf{a}| = 2$、$|\mathbf{b}| = 3$ 且 $|\mathbf{c}| = 4$。求 $\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a}$ 的值。

1. Expand the cross product on the left hand side of the equation.

$(\mathbf{a}+\mathbf{b})\times(\mathbf{a}+\mathbf{c}) = \mathbf{a}\times\mathbf{a} + \mathbf{a}\times\mathbf{c} + \mathbf{b}\times\mathbf{a} + \mathbf{b}\times\mathbf{c}$

2. Simplify the equation using the property $\mathbf{a} \times \mathbf{a} = \mathbf{0}$.

$\mathbf{0} + \mathbf{a}\times\mathbf{c} + \mathbf{b}\times\mathbf{a} + \mathbf{b}\times\mathbf{c} = \mathbf{b}\times\mathbf{c}$
$\mathbf{a}\times\mathbf{c} + \mathbf{b}\times\mathbf{a} = \mathbf{0}$

3. Rearrange and factorise the expression using anti-commutativity of the cross product.

Since $\mathbf{b}\times\mathbf{a} = -\mathbf{a}\times\mathbf{b}$:
$\mathbf{a}\times\mathbf{c} - \mathbf{a}\times\mathbf{b} = \mathbf{0}$
$\mathbf{a}\times(\mathbf{c} - \mathbf{b}) = \mathbf{0}$

4. Deduce the relationship between the vectors.

We are given that $\mathbf{a}$ is a non-zero vector, and $\mathbf{b} \ne \mathbf{c}$ implies $\mathbf{c} - \mathbf{b} \ne \mathbf{0}$.
Since the cross product of two non-zero vectors is the zero vector, they must be parallel.
Therefore, $\mathbf{a}$ is parallel to $\mathbf{c} - \mathbf{b}$.

1. Take the dot product of the given vector sum with itself.

$(\mathbf{a} + \mathbf{b} + \mathbf{c}) \cdot (\mathbf{a} + \mathbf{b} + \mathbf{c}) = \mathbf{0} \cdot \mathbf{0} = 0$

2. Expand the dot product using the distributive property and the definition of magnitude.

$|\mathbf{a}|^2 + |\mathbf{b}|^2 + |\mathbf{c}|^2 + 2(\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a}) = 0$

3. Substitute the given magnitudes to find the value of the required expression.

$(2)^2 + (3)^2 + (4)^2 + 2(\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a}) = 0$
$4 + 9 + 16 + 2(\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a}) = 0$
$29 + 2(\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a}) = 0$
$\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a} = -\frac{29}{2} = -14.5$