This problem asks us to prove that for any vectors $a, b, c \in \mathbb{R}^3$:

$(a \times b) \times c + (b \times c) \times a + (c \times a) \times b = 0.$

This expression involves the vector triple product, a useful identity in vector algebra. The vector triple product states:

$\mathbf{(p \times q) \times r} = (\mathbf{r} \cdot \mathbf{p}) \mathbf{q} - (\mathbf{r} \cdot \mathbf{q}) \mathbf{p}.$

We will apply this identity to each term in the equation.

Step 1: Apply the vector triple product identity to each term

First term: $(a \times b) \times c$

Using the vector triple product identity:

$(a \times b) \times c = (c \cdot a)b - (c \cdot b)a.$

Second term: $(b \times c) \times a$

Applying the vector triple product identity again:

$(b \times c) \times a = (a \cdot b)c - (a \cdot c)b.$

Third term: $(c \times a) \times b$

One more time using the triple product identity:

$(c \times a) \times b = (b \cdot c)a - (b \cdot a)c.$

Step 2: Sum all the terms

Now, we add the results from all three terms:

$(a \times b) \times c + (b \times c) \times a + (c \times a) \times b$

Substituting the expressions we found:

[ [(c \cdot a)b - (c \cdot b)a] + [(a \cdot b)c - (a \cdot c)b] + [(b \cdot c)a - (b \cdot a)c]. ]

Step 3: Simplify the expression

Now, group similar terms:

• Terms involving $b$: (c \cdot a)b - (a \cdot c)b = 0 \quad \text{(since c \cdot a = a \cdot c)}.

• Terms involving $a$: -(c \cdot b)a + (b \cdot c)a = 0 \quad \text{(since c \cdot b = b \cdot c)}.

• Terms involving $c$: (a \cdot b)c - (b \cdot a)c = 0 \quad \text{(since a \cdot b = b \cdot a)}.

Step 4: Conclusion

Since all terms cancel out, we have:

$(a \times b) \times c + (b \times c) \times a + (c \times a) \times b = 0.$

Thus, the given vector identity is proved.

Would you like further clarification on any steps or details?

Here are some related questions to explore:

1. What is the geometric interpretation of the vector cross product?
2. How can the vector triple product be derived from basic vector identities?
3. What are other important vector identities in $\mathbb{R}^3$?
4. How does the vector triple product identity generalize to higher dimensions?
5. Can you prove a similar identity for the scalar triple product?

Tip: Remember that vector identities like the triple product are essential tools in simplifying problems involving forces and torques in physics.