the angle between any two vectors is $\frac{\pi}{3}$, we can analyze the vectors $\mathbf{x}$ and $\mathbf{y}$ defined as follows:

$ |\mathbf{a}| = |\mathbf{b}| = |\mathbf{c}| = \sqrt{2} $

First, let's express $\mathbf{x}$:

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

Using the vector triple product identity:

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

Given that the angle between each pair of vectors is $\frac{\pi}{3}$, the dot products are:

$ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos\left(\frac{\pi}{3}\right) = \sqrt{2} \times \sqrt{2} \times \frac{1}{2} = 1 $

Therefore, the expression for $\mathbf{x}$ becomes:

$ \mathbf{x} = 1 \cdot \mathbf{b} - 1 \cdot \mathbf{c} = \mathbf{b} - \mathbf{c} $

Now, consider $\mathbf{y}$:

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

Using the vector triple product identity again:

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

This simplifies similarly to:

$ \mathbf{y} = 1 \cdot \mathbf{c} - 1 \cdot \mathbf{a} = \mathbf{c} - \mathbf{a} $

Hence, the magnitudes of $\mathbf{x}$ and $\mathbf{y}$ are:

$ |\mathbf{x}| = |\mathbf{b} - \mathbf{c}| $

$ |\mathbf{y}| = |\mathbf{c} - \mathbf{a}| $

Since both expressions have the same form, it follows that:

$ |\mathbf{x}| = |\mathbf{y}| $

Vector Algebra

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